3409FGAfga3.iifga3.ii.xmlThe existence theorem and the formal theory of modules3.II
A. Grothendieck.
"Technique de descente et théorèmes d'existence en géométrie algébrique, II: Le théorème d'existence et théorie formelle des modules".
Séminaire Bourbaki 12 (1959–60), Talk no. 195.
(Numdam)
2465fga3.ii-afga3.ii-a.xmlRepresentable and pro-representable functorsAfga3.ii2460fga3.ii-a.1fga3.ii-a.1.xmlRepresentable functorsA.1fga3.ii-a
Let \mathcal {C} be a category.
For all X \in \mathcal {C}, let h_X be the contravariant functor from \mathcal {C} to the category \mathtt {Set} of sets given by
\begin {aligned} h_X \colon \mathcal {C} & \to \mathtt {Set} \\ Y& \mapsto \operatorname {Hom} (Y,X). \end {aligned}
If we have a morphism X \to X' in \mathcal {C}, then this evidently induces a morphism h_X \to h_{X'} of functors;
h_X is a covariant functor in X, i.e. we have defined a canonical covariant functor
h \colon \mathcal {C} \to \operatorname {Hom} ( \mathcal {C}^ \circ , \mathtt {Set} )
from \mathcal {C} to the category of covariant functors from the dual \mathcal {C}^ \circ of \mathcal {C} to the category of sets.
We then recall:
807Propositionfga3.ii-a.1-proposition-1.1fga3.ii-a.1-proposition-1.1.xml1.1fga3.ii-a.1
This functor h is fully faithful;
in other words, for every pair X,X' of objects of \mathcal {C}, the natural map
\operatorname {Hom} (X,X') \to \operatorname {Hom} (h_X,h_{X'})
is bijective.
In particular, if a functor F \in \operatorname {Hom} ( \mathcal {C}^ \circ , \mathtt {Set} ) is isomorphic to a functor of the form h_X, then X is determined up to unique isomorphism.
We then say that the functor F is representable.
The above proposition then implies that the canonical functor h defines an equivalence between the category \mathcal {C} and the full subcategory of \operatorname {Hom} ( \mathcal {C}^ \circ , \mathtt {Set} ) consisting of representable functors.
This fact is the basis of the idea of a "solution of a universal problem", with such a problem always consisting of examining if a given (contravariant, as here, or covariant, in the dual case) functor from \mathcal {C} to \mathtt {Set} is representable.
Note further that, just by the definition of products in a category [Gro1957], the functor h \colon X \mapsto h_X commutes with products whenever they exist (and, more generally, with finite or infinite projective limits, and, in particular, with fibred products, taking "kernels" [], etc., whenever such things exist): we have an isomorphism of functors
h_{X \times X'} \xrightarrow { \sim } h_X \times h_{X'}
whenever X \times X' exists, i.e. we have functorial (in Y) bijections
h_{X \times X'} \xrightarrow { \sim } h_X(Y) \times h_{X'}(Y).
In particular, the data of a morphism
X \times X' \to X''
in \mathcal {C} (i.e. of a "composition law" in \mathcal {C} between X, X', and X'') is equivalent to the data of a morphism h_{X \times X'}=h_X \times h_{X'} \to h_{X''}, i.e. to the data, for all Y \in \mathcal {C}, of a composition law of sets
h_X(Y) \times h_{X'}(Y) \to h_{X''}(Y)
such that, for every morphism Y \to Y' in \mathcal {C}, the system of set maps
h_{X^{(i)}}(Y) \to h_{X^{(i)}}(Y') \qquad \text {for }i=0,1,2
is a morphism for the two composition laws, with respect to Y and Y'.
In this way, we see that the idea of a "\mathcal {C}-group" structure, or a "\mathcal {C}-ring" structure, etc. on an object X of \mathcal {C} can be expressed in the most manageable way (in theory as much as in practice) by saying that, for every Y \in \mathcal {C}, we have a group law (resp. ring law, etc.) in the usual sense on the set h_X(Y), with the maps h_X(Y) \to h_X(Y') corresponding to morphisms Y \to Y' that should be group homomorphisms (resp. ring homomorphisms, etc.).
This is the most intuitive and manageable way of defining, for example, the various classical groups \operatorname {G_a}, \operatorname {G_m}, \operatorname {GL} (n), etc. on a prescheme S over an arbitrary base, and of writing the classical relations between these groups, or of placing a "vector bundle" structure on the affine scheme V( \mathscr { F } ) over S defined by a quasi-coherent sheaf \mathscr { F }, and of defining and studying the many associated flag varieties (Grassmannians, projective bundles), etc.;
the general yoga is purely and simply identifying, using the canonical functor h, the objects of \mathcal {C} with particular contravariant functors (namely, representable functors) from \mathcal {C} to the category of sets.
The usual procedure of reversing the arrows that is necessary, for example, in the case of affine schemes in order to pass from the geometric language to the language of commutative algebra, leads us to dualise the above considerations, and, in particular, to also introduce covariant representable functors \mathcal {C} \to \mathtt {Set}, i.e. those of the form Y \mapsto \operatorname {Hom} (X,Y)=h'_X(Y).
2461fga3.ii-a.2fga3.ii-a.2.xmlPro-representable functors, pro-objectsA.2fga3.ii-a
Let \mathcal {X}=(X_i)_{i \in I} be a projective system of objects of \mathcal {C};
there is a corresponding covariant functor
h'_{ \mathcal {X}} = \varinjlim _i h'_{X_i}
which can be written more explicitly as
h'_{ \mathcal {X}}(Y) = \varinjlim _i h'_{X_i}(Y) = \varinjlim _i \operatorname {Hom} (X_i,Y)
which is a functor from \mathcal {C} to \mathtt {Set}.
A functor from \mathcal {C} to \mathtt {Set} that is isomorphic to a functor of this type with I filtered is said to be pro-representable.
By the previous section, these are exactly the functors that are isomorphic to filtered inductive limits of representable functors.
Let \mathcal {X}'=(X_j)_{j \in J} be another filtered projective system in \mathcal {C} (indexed by another filtered preordered set of indices J).
Then we can easily show that we have a canonical bijection
\operatorname {Hom} (h_{ \mathcal {X}'},h_{ \mathcal {X}}) = \varprojlim _j \varinjlim _i \operatorname {Hom} (X_i,X'_j)
(generalising ).
This leads to introducing the category \operatorname {Pro} ( \mathcal {C}) of pro-objects of \mathcal {C}, whose objects are projective systems of objects of \mathcal {C} (indexed by arbitrary filtered preordered sets of indices), and whose morphisms between objects \mathcal {X}=(X_i)_{i \in I} and \mathcal {X}'=(X_j)_{j \in J} are given by
\operatorname {Pro} \operatorname {Hom} ( \mathcal {X}, \mathcal {X}') = \varprojlim _j \varinjlim _i \operatorname {Hom} (X_i,X'_j),
with the composition of pro-homomorphisms being evident.
By the very construction itself, \mathcal {X} \mapsto h'_{ \mathcal {X}} can be considered as a contravariant functor in \mathcal {X}, establishing an equivalence between the dual category of the category \operatorname {Pro} ( \mathcal {C}) of pro-objects of \mathcal {C} and the category of pro-representable covariant functors from \mathcal {C} to \mathtt {Set}.
Of course, an object X of \mathcal {C} canonically defines a pro-object, denoted again by X, so that \mathcal {C} is equivalent to a full subcategory of \operatorname {Pro} ( \mathcal {C}).
Then, if \mathcal {X}=(X_i)_{i \in I} is an arbitrary pro-object of \mathcal {C}, then (with the above identification) we have that
\mathcal {X} = \varprojlim _i X_i
with the projective limit being taken in \operatorname {Pro} ( \mathcal {C}) (since h_{ \mathcal {X}}= \varinjlim _i h_{X_i}).
We draw attention to the fact that, even if the projective limit of the X_i exists in \mathcal {C}, it will generally not be isomorphic to the projective limit \mathcal {X} in \operatorname {Pro} ( \mathcal {C}), as is already evident in the case where \mathcal {C} is the category of sets.
We note that, by the definition itself, \varprojlim {}_{ \mathcal {C}}X_i=L is defined by the condition that the functor
\varprojlim _i \operatorname {Hom} _{ \mathcal {C}}(Y,X_i) = \operatorname {Hom} _{ \operatorname {Pro} ( \mathcal {C})}(Y, \mathcal {X})
in Y \in \mathcal {C} and with values in \mathtt {Set} be representable via \mathcal {L}, i.e. that it be isomorphic to \operatorname {Hom} _{ \mathcal {C}}(Y, \mathcal {L});
then \lim {}_{ \mathcal {C}}X_i is already defined in terms of the pro-object \mathcal {X}, and, in a precise way, depends functorially on the pro-object \mathcal {X} whenever it is defined;
there is therefore no problem with denoting it by \lim {}_{ \mathcal {C}}( \mathcal {X}).
If projective limits in \mathcal {C} always exist, then \lim {}_{ \mathcal {C}}( \mathcal {X}) is a functor from \operatorname {Pro} ( \mathcal {C}) to \mathcal {C}, and there is a canonical homomorphism of functors \lim _ \mathcal {C}( \mathcal {X}) \to \mathcal {X}.
Since every (covariant, say, for simplicity) functor
F \colon \mathcal {C} \to \mathcal {C}'
can be extended in an obvious way to a functor
\operatorname {Pro} (F) \colon \operatorname {Pro} ( \mathcal {C}) \to \operatorname {Pro} ( \mathcal {C}'),
it follows that, if projective limits always exist in \mathcal {C}', then F also canonically defines a composite functor
\overline {F} = \varprojlim {}_{ \mathcal {C}'} \colon \operatorname {Pro} ( \mathcal {C}) \to \mathcal {C}'
sending \mathcal {X}=(X_i)_{i \in I} to \varprojlim {}_{ \mathcal {C}'}F(X_i).
A pro-object \mathcal {X} is said to be a strict pro-object if it is isomorphic to a pro-object (X_i)_{i \in I}, where the transition morphisms X_i \to X_j are epimorphisms;
a functor defined by such an object is said to be strictly pro-representable.
We can thus further demand that I be a filtered ordered set, and that every epimorphism X_i \to X' be equivalent to an epimorphism X_i \to X_j for some suitable j \in I (uniquely determined by this condition).
Under these conditions, the projective system (X_i)_{i \in I} is determined up to unique isomorphism (in the usual sense of isomorphisms of projective systems).
It thus follows that the projective limit of a projective system \mathcal {X}^{( \alpha )} of strict pro-objects always exists in \operatorname {Pro} ( \mathcal {C}), and that, with the above notation of F and \overline {F}, we have that
\overline {F} \varprojlim _ \alpha \mathcal {X}^{( \alpha )} = \varprojlim _ \alpha {}_{ \mathcal {C}'}F(X^{( \alpha )}).
In particular, if every pro-object of \mathcal {C} is strict (cf. the previous section), then the extended functor \overline {F} commutes with projective limits.
2462fga3.ii-a.3fga3.ii-a.3.xmlCharacterisation of pro-representable functorsA.3fga3.ii-a
Let \mathcal {C} and \mathcal {C}' be categories in which all finite projective limits (i.e. limits over finite, not necessarily filtered, preordered sets) exist, or, equivalently, in which finite products and finite fibred products exist (which implies, in particular, the exists of a "right-unit object" e such that \operatorname {Hom} (X,e) consists of only on element for all X).
Let F be a covariant functor from \mathcal {C} to \mathcal {C}'.
Then the following conditions are equivalent:
F commutes with finite projective limits;
F commutes with finite products and with finite fibred products;
F commutes with finite products, and, for every exact diagram
X \to X' \rightrightarrows X''
in \mathcal {C} (cf. FGA 3.I, A, Definition 2.1), the image of the diagram under F
F(X) \to F(X') \rightrightarrows F(X'')
is exact.
We then say that F is left exact.
In what follows, we assume that finite projective limits always exist in \mathcal {C}.
It is then immediate from the definitions that a representable functor is left exact, and, by taking the limit, that a pro-representable functor is left exact.
To obtain a converse, let
F \colon \mathcal {C} \to \mathtt {Set}
be a covariant functor, and let X \in \mathcal {C} and \xi \in F(X).
We say that \xi (or the pair (X, \xi )) is minimal if, for all X' \in \mathcal {C} and all \xi ' \in F(X'), and for every strict monomorphism (cf. FGA 3.I, §A.2) u \colon X' \to X such that \xi =F(u)( \xi '), u is an isomorphism.
We also say that a pair (X, \xi ) dominates (X'', \xi '') (where \xi \in F(X) and \xi '' \in F(X'')) if there exists a morphism v \colon X \to X'' such that \xi ''=F(v)( \xi );
if \xi is minimal, and if F is left exact, then this morphism v is unique;
if \xi '' is minimal, then v is surjective.
From this we easily deduce the following proposition:
1505Propositionfga3.ii-a.3-proposition-3.1fga3.ii-a.3-proposition-3.1.xml3.1fga3.ii-a.3
For F to be strictly pro-representable, it is necessary and sufficient that it satisfy the following two conditions:
F is left exact; and
every pair (X, \xi ), with \xi \in F(X), is dominated by some minimal pair.
This second condition is trivial if every object of \mathcal {C} is Artinian (by taking a sub-object X' of X that is minimal amongst those for which there exists some \xi ' \in F(X') such that \xi is the image of \xi ').
Whence:
1506Corollaryfga3.ii-a.3-proposition-3.1-corollaryfga3.ii-a.3-proposition-3.1-corollary.xmlfga3.ii-a.3
Let \mathcal {C} be a category whose objects are all Artinian and in which all finite projective limits exist.
Then the pro-representable functors from \mathcal {C} to \mathtt {Set} are exactly the left exact functors, and they are in fact strictly pro-representable.
This last fact also implies that every pro-object of \mathcal {C} is then strict.
2463fga3.ii-a.4fga3.ii-a.4.xmlExample: groups of Galois type, pro-algebraic groupsA.4fga3.ii-a
If \mathcal {C} is the category of ordinary finite groups, then \operatorname {Pro} ( \mathcal {C}) is equivalent to the category of totally disconnected compact topological groups.
([Trans.] Here the word "Hausdorff" is implicit.)
It is groups of this type, and their generalisations, obtained by replacing ordinary finite groups with schemes of finite groups over a given base prescheme (for example, finite algebraic groups over a field k), that serve as fundamental groups, homotopy groups, and absolute and relative homology groups for preschemes.
In all these examples, the corollary to applies, and it is indeed by the associated functor that the \pi _1 should be defined [].
It is the same if we start with the category of algebraic or quasi-algebraic groups over a field (or, more generally, over a Noetherian prescheme): we recover the "pro-algebraic groups" of Serre [Ser1958].
2464fga3.ii-a.5fga3.ii-a.5.xmlExample: "formal varieties"A.5fga3.ii-a
Let \Lambda be a Noetherian ring, \mathcal {C} the category of \Lambda-algebras that are Artinian modules of finite type over \Lambda (or, more concisely, Artinian \Lambda-algebras).
The conditions of the corollary to
are then satisfied.
Here, the category \operatorname {Pro} ( \mathcal {C}) is equivalent to the category of topological algebras O over \Lambda that are isomorphic to topological projective limits
O = \varprojlim O_i
of algebras O_i \in \mathcal {C}, i.e. those whose topology is linear, separated, and complete, and such that, for every open ideal \mathfrak {J}_i of O, the algebra O/ \mathfrak {J}_i is an Artinian algebra over \Lambda.
The functor \mathcal {C} \to \mathtt {Set} associated to such an algebra is exactly
\begin {aligned} F(A) &= h'_{O}(A) \\ &= \{ \text {continuous homomorphisms of topological } \Lambda \text {-algebras }O \to A \} \\ &= \varinjlim _i \operatorname {Hom} _{ \Lambda \text {-algebras}}(O_i,A). \end {aligned}
Note also that the category \mathcal {C} is essentially the product of analogous categories, corresponding to the local rings that are the completions of the \Lambda _{ \mathfrak {m}} for the maximal ideals \mathfrak {m} of \Lambda;
we can thus, if so desired, restrict to the case where A is such a complete local ring.
In any case, O decomposes canonically as the topological product of its local components, which correspond to the "points" of the formal scheme [] defined by O.
Such a point is defined by an object \xi of some F(K), where K \in \mathcal {C} is a field (for example, the residue field of the local component in question), and where two pairs ( \xi ,K) and ( \xi ',K') define the same point if and only if they are both dominated by the same ( \xi '',K''), or if they both dominate the same ( \xi ''',K''').
(If the { \Lambda }/ \mathfrak {m} are algebraically closed, then it suffices to take the set given by the sum of the F({ \Lambda }/ \mathfrak {m})).
It is important to give conditions that ensure that the local component O_ \xi of O corresponding to some \xi \in F(K) be a Noetherian ring.
If \Lambda is a complete local ring (Noetherian, we recall), then it is equivalent to say that O_ \xi is isomorphic to a quotient ring of a formal series ring \Lambda [{[t_1, \ldots ,t_n]}].
To give such a criterion, we introduce (for every ring A) the A-algebra I_A of "dual numbers" of A, defined by
I_A = A[t]/t^2A[t].
Let \varepsilon \colon I_A \to A be the augmentation homomorphism, which defines (if A \in \mathcal {C}) a map
F( \varepsilon ) \colon F(I_A) \to F(A).
Using the fact that F is left exact, we intrinsically define the structure of an A-module on the subset
F(I_A, \xi ) = F( \xi )^{-1}( \xi ) \subset F(I_A)
consisting of the \xi ' \in F(I_A) that are "reducible along \xi";
using the explicit form of F in terms of O, we find that this K-module can be identified with \operatorname {Hom} _ \Lambda ( \mathfrak {m}_{ \xi }/ \mathfrak {m}_ \xi ^2,A), where m_ \xi is the kernel of the homomorphism \xi \colon O \to A, i.e. if A is a field, then the maximal ideal of the local component O_ \xi of O.
From this, we immediately deduce the following proposition:
524Propositionfga3.ii-a.5-proposition-5.1fga3.ii-a.5-proposition-5.1.xml5.1fga3.ii-a.5
Let \xi \in F(K), where K \in \mathcal {C} is a field.
For the corresponding local component O_ \xi of O to be a Noetherian ring, it is necessary and sufficient that the set F(I_K, \xi ) of elements of F(I_K) that are reducible along \xi be a vector space of finite dimension over K.
Under these conditions, we have a canonical isomorphism
F(I_K, \xi ) = \operatorname {Hom} ( \mathfrak {m}_{ \xi }/ \mathfrak {m}_ \xi ^2+ \mathfrak {n}_ \xi \mathscr { O } _ \xi , K)
(where \mathfrak {n}_ \xi is the maximal ideal of \Lambda given by the kernel of the homomorphism \Lambda \to K), and so, in particular, the dimension of the K-vector space F(I_K, \xi ) is equal to the dimension of the vector space \mathfrak {m}_{ \xi }/ \mathfrak {m}_ \xi ^2 over the field O_{ \xi }/ \mathfrak {m}_ \xi =K( \xi ).
[Comp.]
The formula given above is only correct when \Lambda is a field; in the general case, we must replace \mathfrak {m}_{ \xi }/ \mathfrak {m}_ \xi ^2 with the quotient of this space by the image of \mathfrak {n}_{ \xi }/ \mathfrak {n}_ \xi ^2, where \mathfrak {n} is the maximal ideal of \Lambda.
Suppose that O_ \xi is Noetherian, and suppose, for notational simplicity, that \Lambda is complete and local, and that O=O_ \xi.
([Comp.] The following definition is correct only when the residue extension k'/k is separable; for the general case, see [Gro1960b, III, 1.1].)
We say that O is simple over \Lambda if O is a finite and étale algebra over the completion algebra of the localisation of \Lambda [t_1, \ldots ,t_n] at one of its maximal ideals that induces the maximal ideal of \Lambda;
if the residue extension of O over \Lambda is trivial (for example, if the residue field of \Lambda is algebraically closed), then this is equivalent to saying that O itself is isomorphic to such a formal series algebra.
Finally, if we no longer necessarily suppose that O is Noetherian, then we again say that O is simple over \Lambda if O is isomorphic to a topological projective limit of quotient \Lambda-algebras that are Noetherian and \Lambda-simple in the above sense.
We can immediately generalise to the case where \Lambda and O are no longer assumed to be local.
With this, we have the following proposition:
525Propositionfga3.ii-a.5-proposition-5.2fga3.ii-a.5-proposition-5.2.xml5.2fga3.ii-a.5
For O to be simple over \Lambda, it is necessary and sufficient that the associated functor F send epimorphisms to epimorphisms.
If this is the case, then this implies that, for every surjective homomorphism A \to A' in \mathcal {C}, the morphism F(A) \to F(A') is also surjective.
Of course, it suffices to verify this condition in the case where A is local, and (proceeding step-by-step) where the ideal of A given by the kernel of A \to A' is annihilated by the maximal ideal of A.
This leads, in practice, to verifying that a certain obstruction, linked to infinitesimal invariants of the situation that give us a functor F, is null;
this is a problem of a cohomological nature.
To finish, we say some words, in the above context, about rings of definition.
Let F still be a functor from \mathcal {C} to \mathtt {Set}, assumed to be pro-representable via a topological \Lambda-algebra O.
Then, for every A \in \mathcal {C} and every \xi \in F(A), there exists a smallest subring A' of A such that \xi is the image of an element \xi ' of F(A') (which is then uniquely determined):
indeed, it suffices to think of \xi as a homomorphism from O to A, and to take A' to be the image of O under this \xi.
We then say that A' is the ring of definition of the object \xi \in F(A).
If u \colon A \to B is an algebra homomorphism, and if \eta =F(u)( \xi ), then the ring of definition of \eta is the image under u of the ring of definition of \xi.
If we start with a functor F from \mathcal {C} to \mathtt {Set}, then the existence of rings of definition, along with their properties that we have just discussed, is more or less equivalent to the condition that F be pro-representable;
that is, they are usually far from being trivial.
2466fga3.ii-bfga3.ii-b.xmlThe two existence theoremsBfga3.ii
Keeping the notation of , and, given a covariant functor
F \colon \mathcal {C} \to \mathtt {Set} ,
we wish to find manageable criteria for F to be pro-representable, i.e. expressible via a \Lambda-algebra O as above.
By the corollary of §A, Proposition 3.1, to ensure this, it is necessary and sufficient that F be left exact.
In the current state of the technique of descent (cf. the questions asked in FGA 3.I, §A.2.c), this criterion is not directly verifiable, in this form, in the most important cases, and we need criteria that seem less demanding.
1549Theoremfga3.ii-b-theorem-1fga3.ii-b-theorem-1.xml1fga3.ii-b
For the functor F to be pro-representable, it is necessary and sufficient that it satisfy the two following conditions:
F commutes with finite products;
for every algebra A \in \mathcal {C} and every homomorphism A \to A' in \mathcal {C} such that the diagram
A \to A' \rightrightarrows A' \otimes _A A'
is exact (cf. FGA 3.I, §A, Definition 1.2), the diagram
F(A) \to F(A') \rightrightarrows F(A' \otimes _A A')
is also exact.
Furthermore, it suffices to verify condition (ii) in the case where A is local, and when, further, we are in one of the two following cases:
A is a free module over A;
the quotient module A'/A is an A-module of length 1.
1441Prooffga3.ii-b-theorem-1
The proof of this theorem is rather delicate, and cannot be sketched here.
We content ourselves with pointing out that it relies essentially on a study of equivalence relations (in the sense of categories) in the spectrum of an Artinian algebra (the study of which poses even more problems, whose solutions seems essential for the further development of the theory).
In applications, the verification of condition (i) is always trivial.
The verification of condition (ii) splits into two cases: case (a), where A' is a free A-module, can be dealt with using the technique of descent by flat morphisms (cf. FGA 1, Theorems 1, 2, and 3), which offers no difficulty;
to deal with case (b), we will use the following result:
1550Theoremfga3.ii-b-theorem-2fga3.ii-b-theorem-2.xml2fga3.ii-b
Let A be a local Artinian ring with maximal ideal \mathfrak {m}, and let A' be an A-algebra containing A, and such that \mathfrak {m}A' \subset A, and A \to A' \rightrightarrows A' \otimes _A A' is exact (which is the case, in particular, if A'/A is an A-module of length 1).
Let \mathcal {F} be the fibred category (cf. FGA 3.I, §A, Definition 1.1) of quasi-coherent sheaves that are flat over varying preschemes.
Then the morphism \operatorname {Spec} (A') \to \operatorname {Spec} (A) is a strict \mathcal {F}-descent morphism (cf. FGA 3.I, §A, Definition 1.7).
1428Prooffga3.ii-b-theorem-2
We prove this by first proving that
\operatorname {H} ^i(A'/A, \operatorname {G_a} ) = 0 \qquad \text {for }i \geqslant1
(cf. FGA 3.I, §A.4.e), with the hypothesis that \mathfrak {m}A' \subset A allowing us to easily reduce to the case where A is a field (namely A/ \mathfrak {m}).
We can then apply the equivalences described in FGA 3.I, §A.4.e.
In other words, the data of a flat A-module M is completely equivalent to the data of a flat A'-module M' endowed with an (A' \otimes _A A')-isomorphism from M' \otimes _A A' to A' \otimes _A M' satisfying the usual transitivity condition for a descent data (loc. cit.).
2472fga3.ii-cfga3.ii-c.xmlApplications to some particular casesCfga3.ii2467fga3.ii-c.1fga3.ii-c.1.xmlGeneral remarks on functors represented by preschemesC.1fga3.ii-c
Let S be a locally Noetherian prescheme.
A prescheme X over S is said to be locally of finite type over S if, for all x \in X that project to y \in Y, there exists an affine neighbourhood of y of ring A, and an affine neighbourhood of x (over the aforementioned affine neighbourhood of y) of ring B, such that B is an A-algebra of finite type.
There are many important examples of preschemes locally of finite type over S, that are not of finite type over S, given by solutions of classical universal problems;
thus it is important to be able to consider the Picard scheme of a curve as a union of infinitely-many connected components (that we must avoid confusing with the connected component of the identity element, i.e. the "Picard variety").
It is thus sometimes useful to place ourselves in the category \mathcal {C} of preschemes locally of finite type over S, in order to study the question of representability of a contravariant functor F.
The main goal of these articles is to develop a general technique that allows us to recognise when such a functor F is representable, and to study the properties of the corresponding S-prescheme X by means of the properties of F.
We note in passing that, in this study, we find non-pathological examples of preschemes over S that are not separated over S, notably as "Picard preschemes" of excellent S-schemes;
we must thus refrain from banishing preschemes that are not schemes from algebraic geometry.
Let X be a prescheme locally of finite type over S, and let
F \colon Y \mapsto \operatorname {Hom} _S(Y,X)
be the associated contravariant functor.
We can consider the restriction F_0 of F to the subcategory \mathcal {C}_0 of \mathcal {C} consisting of preschemes Y over S that are Artinian and finite over S:
if S= \operatorname {Spec} ( \Lambda ), then \mathcal {C}_0 is the category dual to the category of Artinian \Gamma-algebras considered in .
If Y= \operatorname {Spec} (A), where A is a local Artinian ring, then Y consists of a single point y living above a closed point s of S, and an S-homomorphism from Y to X (i.e. an element of F(Y)) is defined by the data of a point x \in X over s, along with an \mathscr { O } _s-homomorphism from \mathscr { O } _x to A.
If there exists such a homomorphism, then x is necessarily a closed point of X (since its residue field is algebraic over the residue field of s).
This thus shows that the restriction F_0 of F to "Artinian Y-algebras" is pro-representable, and is represented by the topological Y-algebra whose local components are the completions \widehat { \mathscr { O } _x} of the local rings of X at the points x of X that are closed and live above closed points of Y.
This shows that only knowing F_0 gives precise information about the structure of X (that is, the structure of the completions of its local rings at the aforementioned points).
We note that, even in the case where S is the spectrum of an algebraically closed field, it is only thanks to the systematic consideration of "varieties" Y such that \mathscr { O } _Y may admit nilpotent elements (and, in particular, working with the spectra of local Artinian rings) that we can arrive at the "good formulation" of classical universal problems, and understand the "infinitesimal" aspect.
If we start with a given functor F, and we want to know whether or not it is representable, then studying the functor F_0 (using and ) will give quasi-complete hints;
either, as is often the case (by simply testing, for example, the nature of the sets F(I_K, \xi ) and their functorial behaviour, cf. ), F_0 is already not pro-representable (which explains the failure of attempts made up until now to define varieties of modules in a reasonably natural way for the classification of vector bundles of rank >1);
or we might be able to show that F_0 is indeed representable, but that that vector spaces F(I_K, \xi ) are not of finite dimension, in which case we must be content with the "formal" solution;
or it could be the case that F_0 is indeed representable by a product of complete Noetherian local rings, which gives very strong assumptions for F itself to be representable, and, combined with the analogous properties (but of a more global nature) that we will later develop, will in all likelihood suffice to imply that it is indeed so.
Finally, we come across interesting geometric problems (see and below) where we have only the functor F_0 (not coming from any "global" functor F), and where we will consider ourselves content if we can associate to it a "formal scheme of modules".
To finish these generalities, we note how the theory of schemes explains some apparent anomalies, such as the Igusa surface V whose "Picard variety" P consists of a single point, and for which, however, \operatorname {H} ^1(V, \mathscr { O } _V) \neq0;
in this case, P is a non-trivial "purely infinitesimal" group, i.e. defined by a local algebra \mathscr { O } of finite rank over the base field k and endowed with a diagonal map corresponding to the multiplicative structure of P;
if \mathfrak {m} is the maximal ideal of \mathscr { O }, then the dual of \mathfrak {m}/ \mathfrak {m}^2 is canonically isomorphic to \operatorname {H} ^1(V, \mathscr { O } _V) (cf. below).
It is only when the Picard group is an algebraic group in the classical sense (i.e. simple over the base field k) that the dimension of \operatorname {H} ^1(V, \mathscr { O } _V) (which is always equal to that of \mathfrak {m}/ \mathfrak {m}^2) is equal to that of the Picard group.
2468fga3.ii-c.2fga3.ii-c.2.xmlThe schemes \underline { \operatorname {Hom}}_S(X,Y), \prod _{X/S}Z, \underline { \operatorname {Aut}}(X), etc.C.2fga3.ii-c
Let X and Y be preschemes over S;
for every prescheme T over S, let X_T=X \times _S T and Y_T=Y \times _S T, and consider the set
F(T) = \operatorname {Hom} _T(X_T,Y_T) = \operatorname {Hom} _S(X_T,Y) = \operatorname {Hom} _S(X \times _S T,Y)
as a contravariant functor in T.
If it is representable, then we denote by \underline { \operatorname {Hom} } _S(X,Y) the prescheme over S that represents it, and we then have a functorial isomorphism
\operatorname {Hom} _S(T, \underline { \operatorname {Hom} } _S(X,Y)) \xrightarrow { \sim } \operatorname {Hom} _S(T \times _S X,Y).
There are variants of this universal problem, the solutions to which can be summarised as follows: a prescheme of S-automorphisms of an S-prescheme X (which will be a prescheme in groups), a prescheme of S-homomorphisms from an S-prescheme in groups to another (which will be a prescheme in commutative groups if the latter scheme in groups is commutative), etc.
We can also generalise the definition of \underline { \operatorname {Hom} } _S(X,Y) by considering a prescheme Z over the prescheme X over S, and the functor
F(T) = \operatorname {Hom} _{X_T}(X_T,Z_T)
(the set of "sections" of Z_T over X_T);
if this functor is representable, then the S-prescheme that represents it will be denoted by \Pi _{X/S}Z, and we will thus have, by definition, a functorial isomorphism
\operatorname {Hom} _S(T, \Pi _{X/S}Z) = \operatorname {Hom} _{X_T}(X_T,Z_T).
Setting Z=Y \times _S X, we recover \underline { \operatorname {Hom} } _S(X,Y).
From these definitions follows a formula for the new preschemes thus introduced that is as trivial as it is useful, that we will not give here (given that it holds in every category where products and fibred products exist).
More serious is the question of existence of schemes of the type \underline { \operatorname {Hom} } _S(X,Y).
We note first of all that, for fixed X, \underline { \operatorname {Hom} } _S(X,Y) (resp. \Pi _{X/S}Z) can only exist for all Y over S (resp. for all Z over X) if X is flat over S.
Furthermore, we can convince ourselves that it is not reasonable to expect the existence of a solution, for general enough Y, except in the case where X is further proper over S.
It seems, however, that these conditions are sufficient for the existence of \underline { \operatorname {Hom} } _S(X,Y) and \Pi _{X/S}Z, with the condition that, if necessary, we make some sort of "quasi-projective" hypothesis on Y/S (resp. Z/X);
this is what we can verify anyway in numerous cases (for example, when Y is affine over S, or, by direct elementary constructions, when X is finite over S).
Then and give:
1702Propositionfga3.ii-c.2-proposition-2.1fga3.ii-c.2-proposition-2.1.xml2.1fga3.ii-c.2
Let \Lambda be a Noetherian ring, and X and Y arbitrary preschemes over \Lambda.
Consider the functor
F(A) = \operatorname {Hom} _A(X_A,Y_A)
on the category \mathcal {C}_0 of Artinian \Lambda-algebras.
If X is flat over \Lambda, then this functor is pro-representable.
Furthermore, we can show that, for all A \in \mathcal {C}_0 and all \xi \in F(A), we have a canonical isomorphism
F(I_A, \xi ) = \operatorname {H} ^1 \Big (X_A, \underline { \operatorname {Hom} } _{ \mathscr { O } _{X_A}} \big ( \xi ^*( \Omega _{Y_A/A}^1), \mathscr { O } _{X_A} \big ) \Big )
where \Omega _{Y_A/A}^1 is the sheaf of Kähler 1-differentials of Y_A with respect to A.
Taking A to be a field, we find, using §A, Proposition 5.1 and the finiteness theorem from , the following corollary:
1703Corollaryfga3.ii-c.2-proposition-2.1-corollaryfga3.ii-c.2-proposition-2.1-corollary.xmlfga3.ii-c.2
Suppose that X is flat and proper over S, and that Y is of finite type over S.
Then F is pro-representable, and the local components of the corresponding topological \Lambda-algebra are Noetherian rings.
1704Remarksfga3.ii-c.2-remarksfga3.ii-c.2-remarks.xmlfga3.ii-c.2
The problems considered in this section, and many others, having been generally studied, in the framework of classical algebraic geometry, via the "Chow coordinates" of cycles in projective space, allow us to consider these cycles as points of suitable projective varieties.
This procedure, and, more generally, the use of Chow coordinates, seems irredeemably insufficient from the point of view of schemes, since it destroys the nilpotent elements in the parameterised varieties, and, in particular, do not lend themselves to a satisfying study of infinitesimal variations of cycles (without taking its non-intrinsic nature, linked to the projective space, into account).
The language of Chow coordinates has sadly been the only one used by many algebraic geometers for the study of families of varieties or families of cycles, which seems to have been a serious obstacle to the understanding of these notions, despite its certain technical interest (probably temporary).
If we wish to obtain the analogue of Chow varieties in the theory of schemes, we are led to the following universal problem:
let X be a prescheme over S, and, for every prescheme T over S, consider the set F(T) of closed sub-preschemes of X_T=X \times _S T that are flat over T; we want to represent this functor in T via some prescheme over S.
More generally, we can start with a quasi-coherent sheaf \mathscr { G } on X, and take F(T) to be the set of quotient sheaves of \mathscr { G } _T that are flat over T.
It seems that there exists a solution to this problem, with a scheme C that is locally of finite type over S, if X is proper over S, if S is locally Noetherian, and if F is furthermore coherent.
In any case, supposing only that S is locally Noetherian, the restriction of F to "Artinian S-algebras" is pro-representable, and, if, furthermore, X is proper over S, and F is coherent, then the local components of the corresponding topological ring \mathscr { O } are Noetherian.
Of course, even after having proven the existence of the "Chow scheme" of X over S, it remains to find a decomposition of it into disjoint open subsets C_i (corresponding to fixing continuous invariants, such as degree and dimension of the cycles that we vary) over S, as well as to make precise the relations between this scheme with the classical Chow varieties, and to make precise when a C_i is projective (or at least quasi-projective) over S.
1705Remarkfga3.ii-c.2-remarkfga3.ii-c.2-remark.xmlfga3.ii-c.2[Comp.]
The problems described here are completely resolved in the projective case by "Hilbert schemes" (cf. ).
Examples by Nagata and Hironaka show, however, that the functors described are not necessarily representable if we do not make the projective hypothesis, even if we restrict to the classification of subvarieties of dimension 0 of a complete non-singular variety of dimension 3;
this is linked to the fact that the symmetric square of such a variety does not necessarily exist.
2469fga3.ii-c.3fga3.ii-c.3.xmlPicard schemesC.3fga3.ii-c[Comp.]
For a more complete study, see .
Let f \colon X \to S be an S-prescheme, and consider the multiplicative sheaf \mathscr { O } _X^ \times of units of the structure sheaf of X, along with the group
P(X/S) = \operatorname {H} ^0(S, \operatorname {R} ^1f_*( \mathscr { O } _X^ \times )),
called the relative Picard group of X/S.
An element of this group is thus defined by giving an open cover (U_i) of S, along with an invertible sheaf \mathscr { L } _i on each f^{-1}(U_i), such that \mathscr { L } _i|f^{-1}(U_i \cap U_j) is isomorphic to \mathscr { L } _j|f^{-1}(U_i \cap U_j) for all i,j, or, at least locally over U_i \cap U_j (i.e. these two sheaves are "equivalent" in the sense of FGA 3.I, §B.4).
If X/S admits a section, then P(X/S) is exactly the set of classes of invertible sheaves on X/S up to "equivalence" (loc. cit.).
We now set, for all T over S,
F(T) = P(X_T/T)
which gives a covariant functor in T, that we call the Picard functor of X/S;
if this functor is representable, then the prescheme over S that represents it is called the Picard prescheme of X/S, and denoted by \mathscr { P } (X/S).
In this case, we then have an isomorphism of functors:
\operatorname {Hom} _S(T, \mathscr { P } (X/S)) \xrightarrow { \sim } P(X_T/T).
Taking the Picard prescheme is compatible with extension of the base, and, in particular, the Picard preschemes of the fibres of X over S (which are preschemes over the residue fields K(s) of the s \in S) are the fibres of \mathscr { P } (X/S).
Of course, since P(X_T/T)=F(T) is a commutative group, the Picard preschemes are preschemes in groups.
Note as well that the generalised Jacobians of Rosenlicht are exactly the connected components of the identity in the Picard schemes of complete curves (possibly with singularities), which should make most of their properties clear (once their existence has been proven).
1758Remarkfga3.ii-c.3-remarkfga3.ii-c.3-remark.xmlfga3.ii-c.3
The definition adopted here is only reasonable when every point of Y admits an open neighbourhood U over which X admits a section.
In the general case, it is necessary to slightly modify the definition of the Picard functor in order to still obtain an existence theorem.
Here, the plausible existence conditions for a Picard prescheme are the following: X is proper and flat over S; f_*( \mathscr { O } _X)= \mathscr { O } _S; and X locally admits a section over S.
This condition naturally arises in the application of the technique of descent, in eliminating the automorphisms of an invertible sheaf \mathscr { L } on X by endowing them with a marked section over the section s (FGA 3.I, §B.4).
Notably, we find the following:
1759Propositionfga3.ii-c.3-proposition-3.1fga3.ii-c.3-proposition-3.1.xml3.1fga3.ii-c.3
Suppose that X is flat over S= \operatorname {Spec} ( \Lambda ), where \Lambda is Noetherian, and suppose that, for all T of finite type over S, we have {f_T}_*( \mathscr { O } _{X_T})= \mathscr { O } _T (if f is proper and separable and has separable fibres, or if S is the spectrum of a field, then it follows from Künneth that the latter condition is equivalent to f_*( \mathscr { O } _X)= \mathscr { O } _S).
Then the Picard functor of X/S on the category of Artinian \Lambda-algebras is pro-representable.
Furthermore, we then have
F(I_A, \xi ) = \operatorname {H} ^1(X_A, \mathscr { O } _{X_A}),
and, in particular:
1760Corollaryfga3.ii-c.3-proposition-3.1-corollaryfga3.ii-c.3-proposition-3.1-corollary.xmlfga3.ii-c.3
If X is proper over S, then the local components of the topological \Lambda-algebra corresponding to the Picard functor are Noetherian.
1761Remarksfga3.ii-c.3-remarks-ifga3.ii-c.3-remarks-i.xmlfga3.ii-c.3
We can generalise the definitions and results from this section to the classification of principal bundles on X, with structure group G being a scheme in groups over S that is affine and flat over S, and also commutative.
In the case where G would not be commutative, and thus where the adjoint bundle in groups of a principal bundle (whose sections of the automorphisms of the principal bundle) would no longer be trivial, no longer holds true as it is stated.
We can, however, modify the universal problem in such a way that we again obtain a solution (at least, for now, in formal geometry).
The golden rule to remember, in the context of the current section and in the following, and every time we are looking for "schemes of modules" for classes of objects that are only defined up to isomorphism, is always the following: eliminate the possible automorphisms of the objects that we want to classify, by introducing, if necessary, auxiliary structures (points or elements of marked sections, fixing differential forms, etc.) that we take to be insignificant enough that we do not substantially modify the initial problem.1762Remarksfga3.ii-c.3-remarks-iifga3.ii-c.3-remarks-ii.xmlfga3.ii-c.3[Comp.]
I have recently shown that the formal scheme of modules for an abelian variety over a field is indeed simple over the Witt ring, or, in other words, that every abelian scheme X over a local Artinian ring that is the quotient of another such scheme Y comes, by reduction, from an abelian scheme over Y.
The proof simply uses the variance properties of the obstruction class of the covering, introduced in FGA II, §6.
Recall also that the schemes of modules for curves of genus g or for polarised abelian schemes have been constructed by Mumford (cf. Séminaire Mumford–Tate, Harvard University (1961–62)).
2470fga3.ii-c.4fga3.ii-c.4.xmlFormal modules of a varietyC.4fga3.ii-c
Let \Lambda be a local Noetherian ring of residue field k (more often than not, \Lambda will be equal to k, or to a Cohen p-ring), and let X_0 be a prescheme over k.
For every local Artinian \Lambda-algebra A, consider the set F(A) of isomorphism classes of A-preschemes X that are flat over A, endowed with an isomorphism
1709Equationfga3.ii-c.4-equation-starfga3.ii-c.4-equation-star.xml*fga3.ii-c.4 X \otimes _A k(A) \xleftarrow { \sim } X_0 \otimes _k k(A) \tag{*}
where k(A) is the residue field of A;
of course, the isomorphisms between such flat A-preschemes should respect the above isomorphism given in the structure.
If A is a (not necessary local) Artinian \Lambda-algebra, with local components A_i, then we take F(A) to be the product of the F(A_i).
Then F becomes a multiplicative functor in A, and we call it the functor of modules for X_0 (and \Lambda).
If this functor is representable, then it has a corresponding local topological \Lambda-algebra O, of residue field k, and the formal spectrum of O is called the formal scheme of modules for X_0 (and \Lambda) (cf. for some details on this).
Here, if we wish to apply the technique of descent, the "finite" automorphisms of X_0 are inoffensive, since they have no influence on the existence of automorphisms (in the precise sense above) of A-preschemes X;
the necessary and sufficient condition, if A is not simply a field, for X to not have any non-trivial A-automorphisms is that
\operatorname {H} ^0(X_0, \mathfrak {G}_{X_0/k}) = 0
where \mathfrak {G}_{X_0/k} denotes the sheaf of k-derivations (i.e. the tangent sheaf) of X_0.
We can easily show (at least, if X_0 is simple over k) that
F(I_A, \xi ) = \operatorname {H} ^1(X_A, \mathfrak {G}_{X_A/A}).
We thus conclude, as per usual:
1710Propositionfga3.ii-c.4-proposition-4.1fga3.ii-c.4-proposition-4.1.xml4.1fga3.ii-c.4
Suppose that \operatorname {H} ^0(X_0, \mathfrak {G}_{X_0/k})=0.
Then the formal scheme of modules for X_0 exists.
If, furthermore, X_0 is proper over k, then the formal scheme of modules is Noetherian.
1711Remarksfga3.ii-c.4-remarksfga3.ii-c.4-remarks.xmlfga3.ii-c.4
If X_0 is not assumed to be simple over k, then F(I_A, \xi ) can be identified with a sub-A-module of
\operatorname {Ext} _{ \mathscr { O } _{P_A}}^1(P_A; \mathscr { I } _{X_A}, \mathscr { O } _{X_A})
where we set P_A=X_A \times _A X_A, where \mathscr { O } _{X_A} is considered as a coherent sheaf on P_A via the diagonal morphism X_A \to P_A, and where \mathscr { I } _{X_A} denotes the coherent sheaf of ideals on P_A defined by the diagonal morphism.
More precisely, an easy globalisation of Hochschild theory shows that the \operatorname {Ext} ^1 above can be identified with the set of classes, up to isomorphism, of sheaves of I_A-algebras \mathscr { O } that are flat over X_A, and endowed with an augmentation isomorphism \mathscr { O } \otimes _{I_A}A \to \mathscr { O } _{X_A} (recall that we set I_A=At/(t^2)).
The submodule F(I_A, \xi ) is that which corresponds to the sheaves of commutative algebras.
The simplicity hypotheses are thus not essential in the theory of modules, as implies.
Recall (loc. cit.) that, in particular, every simple and proper algebraic curve X_0 over k admits a formal scheme of modules that is simple over \Lambda, and of relative dimension equal to 3g-3 if the genus g is \geqslant2, and to g if g=0,1.
These two latter cases no longer figure directly in .
We can, however, recover them in the case of elliptic curves (g=1) thanks to the remarks that will follow.
We can, of course, vary ad libitum by considering systems of schemes over k endowed with various structures.
Suppose, for example, that X_0 is an abelian scheme over k, with a marked origin (i.e. X_0 is considered as a scheme in groups over k), and let F(A) be the set of isomorphism classes of abelian schemes over A (i.e. of schemes in groups that are proper and simple over A) endowed with an isomorphism as in of abelian schemes.
We can show that imposing a multiplicative structure (or even only a "unit section") eliminates the infinitesimal automorphisms, and that there thus exists a formal scheme of modules that corresponds to a complete local Noetherian ring O.
We can also show that, if X is a proper and simple scheme with "absolutely connected" fibres over a locally Noetherian prescheme S, then every multiplicative structure on X that admits a unit section is necessarily associative and commutative (provided that it is associative and commutative on one fibre, and provided that S is connected), and is furthermore uniquely determined by the knowledge of the unit section.
Further, supposing that S is the spectrum of a local Artinian ring A of residue field k, that X is proper over A and endowed with a section s, and finally that X \otimes _A k is endowed with the structure of an abelian scheme over k, admitting the point of X \otimes _A k corresponding to s as the zero element, an easy calculation of obstructions, combined with an argument due to Serre, allows us to prove that there exists on X a multiplicative structure admitting the section s as the unit section.
(From here, using the "existence theorem" of to pass to the case where A is complete local Noetherian, and then the technique of descent from for the general case, we can prove the analogous claim for all locally Noetherian connected S).
This proves that the functor F(A) considered here is isomorphic to the analogous functor defined at the start of this section by abstracting the multiplicative structure on X_0.
It then follows that, in particular, if \mathfrak {m} is the maximal ideal of O, then \mathfrak {m}/ \mathfrak {m}^2 is canonically isomorphic to the dual of \operatorname {H} ^1(X_0, \mathfrak {G}_{X_0/k}), and is thus of dimension n^2, where n= \dim X_0.
It would be very interesting to determine if O is indeed simple over \Lambda, i.e. isomorphic to an algebra of formal series in n^2 variables over \Lambda.
Now §A, Proposition 5.2 allows us to give an equivalent formulation of this problem as an existence problem of abelian schemes that are reducible along a given abelian scheme.
In any case, we see, by a transcendental way, that the answer is affirmative if k is of characteristic 0.
In characteristic p \neq0, it evidently suffices to restrict to the case where \Lambda is the ring of Witt vectors constructed over an algebraically closed field k.
This could be the moment for the "Greenberg functor" to prove its worth...
2471fga3.ii-c.5fga3.ii-c.5.xmlExtension of coveringsC.5fga3.ii-c
Let \mathfrak {X} be a formal Noetherian prescheme [], U an open subset of \mathfrak {X} defined locally by the "non-vanishing" of a section of \mathscr { O } _ \mathfrak {X} that is not a zero divisor (and thus large enough that every section of \mathscr { O } _ \mathfrak {X} over an open subset V that is zero on U \cap V is zero).
([Comp.] It is also necessary to assume that the section defining U is not a zero divisor not only on \mathfrak {X}, but also on every X_n.)
Let \mathfrak {J} be an "ideal of definition" for \mathfrak {X}, and let X_n=( \mathfrak {X}, \mathscr { O } _ \mathfrak {X}/ \mathfrak {J}^{n+1}), which is thus a ordinary Noetherian prescheme.
Then, if \mathfrak {X}' and \mathfrak {X}'' are flat coverings of \mathfrak {X} (i.e. preschemes over \mathfrak {X} defined by sheaves of algebras that are coherent and locally free as sheaves of modules) that are unramified over U, the evident map
\operatorname {Hom} _ \mathfrak {X}( \mathfrak {X}', \mathfrak {X}'') \to \operatorname {Hom} _{X_0}(X'_0,X''_0)
is injective;
in particular, an automorphism of \mathfrak {X}' that induces the identity on X'_0 is the identity.
This allows us to apply the technique of descent to the situation.
We start, in particular, with a flat covering X'_0 of X_0, unramified over U_0, and let G( \mathfrak {X}) be the set of classes, up to isomorphism (inducing the identity on X'_0), of flat coverings \mathfrak {X}' of \mathfrak {X} that induce X'_0 on X_0 (and that are thus necessarily unramified over U).
We similarly define G(V) for every open subset V of \mathfrak {X}, and, more generally, G( \mathfrak {Y}) for every formal prescheme \mathfrak {Y} over \mathfrak {X}.
With this, the results of and imply, first of all, the following results:
If V varies amongst open subset of \mathfrak {X}, then the G(V) form a sheaf on \mathfrak {X}, say \mathscr { G } _ \mathfrak {X}= \mathscr { G }.
The restriction of this sheaf to U is the constant sheaf whose fibres consist of a single element.
More generally, describing the fibres of \mathscr { G } _ \mathfrak {X} is a question of complete local rings, in a precise way:
For all x \in \mathfrak {X}, we have
\mathscr { G } _x = G( \operatorname {Spec} ( \mathscr { O } _{ \mathfrak {X},x})) \subset G( \operatorname {Spec} ( \widehat { \mathscr { O } }_{ \mathfrak {X},x}))
(i.e. isomorphism classes of finite free algebras B over \widehat { \mathscr { O } }_{ \mathfrak {X},x} endowed with an isomorphism from B \otimes _{ \widehat { \mathscr { O } }_{ \mathfrak {X},x}} \mathscr { O } _{X_0,x} to ( \widehat { \mathscr { O } }'_0)_x, where \mathscr { O } '_0 is the sheaf of algebras on X_0 that defines X'_0).
We have a canonical isomorphism \mathscr { G } _ \mathfrak {X}= \varprojlim \mathscr { G } _{ \mathfrak {X}_n}; in other words, for every open subset V of \mathfrak {X}, we have G(V)= \varprojlim G(V_n).
Suppose that \mathfrak {X} comes from an ordinary proper scheme X over a complete local Noetherian ring \Lambda that has ideal of definition \mathfrak {m} by taking the \mathscr { J }-adic completion of \mathscr { O } _X, where \mathscr { J } = \mathfrak {m} \cdot \mathscr { O } _X.
Then G( \mathfrak {X}) is canonically isomorphic to the set of classes of flat coverings of the ordinary scheme X that are "reducible along X'_0".
Figuratively speaking, we can say that (a) and (b) establish the fundamental relations between the local and global aspects of the problem; (c) gives the relations between the "finite" and "infinitesimal" aspects; and finally (d) remembers (under precise conditions) the identity between the "formal" and "algebraic" aspects.
Now suppose that \mathfrak {X} is defined by a local complete Noetherian ring \Lambda, with \mathscr { J } = \mathfrak {m} \cdot \mathscr { O } _X (and so X_0 is a prescheme over { \Lambda }/ \mathfrak {m}).
For every algebra A that is finite over \Lambda, we set
F(A) = G( \mathfrak {X} \times _ \Lambda A).
This is a covariant functor in A, with values in the category of sets, and, by (c), this functor is completely determined by how it acts on Artinian algebras A;
it is equivalent to say either that this functor is pro-representable, i.e. of the form
F(A) = \operatorname {Hom} _{ \text {top. } \Lambda \text {-algebras}}( \mathscr { O } ,A)
where \mathscr { O } is a topological \Lambda algebra of the type described in , or that this is true when we restrict to only Artinian algebras A.
The combination of and then effectively implies:
1722Propositionfga3.ii-c.5-proposition-5.1fga3.ii-c.5-proposition-5.1.xml5.1fga3.ii-c.5
The above functor is pro-representable.
Of course, by (a), if U= \mathfrak {X}, then G( \mathfrak {Y}) consists of a single element for all \mathfrak {Y} over \mathfrak {X}, and the functor F is then not very interesting (we will have \mathscr { O } = \Lambda).
It seems that, in practically every other case, the topological local ring \mathscr { O } is not Noetherian.
Its existence, however, shows, in a striking manner, the "continuous" nature of the set G( \mathfrak {X}) of solutions (corresponding intuitively to the fact that there is a "continuous" choice in the way in which the ramification spreads when we take an extension of X'_0).
We will compare this result with the point of view of J.-P. Serre [@Ser1958] via local class field theory, drawing attention as well to the continuous character of the topological Galois group of the maximal abelian extension of a "geometric" local field, with the dual group (in the sense of Pontrjagin) appearing as an inductive limit of algebraic (or at least quasi-algebraic) groups;
here as well, the classification of extensions is given by infinite-dimensional "varieties".
We can also take, in the above, \mathfrak {X} to be the formal spectrum of a complete local ring (of which \Lambda will be, for example, a Cohen subring), and we might hope that the results of this section can be used in the study of extensions of a local complete ring of dimension >1.
Just as much in the local case as in the global case, they might allow us to formulate precise relations between the phenomena of higher ramification and phenomena in characteristic 0 (approachable via a transcendental way).
In any case, it is the preliminary analysis of that allows us to extend the methods described in for the study of the fundamental group to the "tamely ramified" case, and to resolve, by a transcendental way, the "problem of three points".
To finish, we note that the situation simplifies if X_0 is of dimension 1;
then, by (a) and (b), G( \mathfrak {X}) can be identified with \prod _i G( \operatorname {Spec} ( \mathscr { O } _{ \mathfrak {X},x_i})), where the x_i are the points of X_0 \setminus U:
we can take arbitrary "local" extensions at the ramification points.
Further, if X_0 is normal, then we note that the formal scheme of modules guaranteed by is simple over \operatorname {Spec} ( \Lambda ).
3399Theoremfga3.vi-3-theorem-3.5fga3.vi-3-theorem-3.5.xml3.5fga3.vi-3
Under the conditions of , let s \in S be such that \underline { \operatorname {Pic} } _{X_s/k(s)} is simple over k(s) (or, equivalently, such that \dim \underline { \operatorname {Pic} } _{X_s/k(s)}= \dim \operatorname {H} ^1(X_s, \mathcal {O} _{X_s})).
Then there exists an open neighbourhood U of s such that \underline { \operatorname {Pic} } _{X/S} is simple over S at the points of \underline { \operatorname {Pic} } _{X/S}^0|U, which is thus an open abelian subscheme in \underline { \operatorname {Pic} } _{X/S}|U.
A fortiori, \underline { \operatorname {Pic} } _{X|U/U}^{00} exists.
2175Prooffga3.vi-3-theorem-3.5
We describe the principle of the proof.
The above allows us to reduce to the case where S is the spectrum of an Artinian local ring A, and we argue by induction on the infinitesimal order of A.
We can thus suppose that \underline { \operatorname {Pic} } _{X_n/A_n}^0 is simple over A_n, and reduce to proving that \underline { \operatorname {Pic} } _{X_{n+1}/A_{n+1}}^0 is simple over A_{n+1}.
Note that, for this, it suffices to construct an abelian scheme P_{n+1} over A_{n+1} that extends P_n= \underline { \operatorname {Pic} } _{X_n/A_n}^0, along with an invertible module \mathscr { L } _{n+1} on X_{n+1} \times _{A_{n+1}}P_{n+1} that extends the invertible module \mathscr { L } _n on X_n \times _{A_n}P_n that arises in the definition of the Picard scheme \underline { \operatorname {Pic} } _{X_n/A_n} as the solution to a universal problem.
(N.B. We can suppose that X is endowed with a section over S...).
For this construction, we must use the following key result: every abelian scheme defined over a quotient of an Artinian local ring can be extended (in other words, the absolute "formal scheme of modules" () for an abelian scheme over an algebraically closed field is simple over the ring of Witt vectors over k);
this result can be obtained by using the general formal properties of the obstruction to lifting, and the group operations.
With this result, we start by extending P_n arbitrarily to P_{n+1};
we then find an obstruction to lifting \mathscr { L } _n, found in \operatorname {H} ^2(X_0 \times P_0, \mathcal {O} _{X_0 \times P_0}) \otimes _k V (where V= \mathfrak {m}^{n+1}/ \mathfrak {m}^{n+2}), and more precisely in the subspace \operatorname {H} ^1(X_0, \mathcal {O} _{X_0}) \otimes \operatorname {H} ^1(P_0, \mathcal {O} _{P_0}) \otimes _k V (taking into account the fact that the restriction of \mathscr { L } _n to the two factors X_n and P_n is trivial).
But this latter space is exactly \operatorname {H} ^1(P_0, \mathscr { G } _{P_0/k}) \otimes V, where \mathscr { G } _{P_0/k} is the tangent sheaf to P_0/k, and thus also the space that expresses the indeterminacy that there was in the lifting of P_n to P_{n+1} ().
So we can correct this lifting (in exactly one way, as should be the case) in such a way as to kill the obstruction to lifting \mathscr { L } _n.
3400fga3.ii-cfga3.ii-c.xmlThe existence theorem and the formal theory of modules › Applications to some particular casesCfga3.ii2467fga3.ii-c.1fga3.ii-c.1.xmlGeneral remarks on functors represented by preschemesC.1fga3.ii-c
Let S be a locally Noetherian prescheme.
A prescheme X over S is said to be locally of finite type over S if, for all x \in X that project to y \in Y, there exists an affine neighbourhood of y of ring A, and an affine neighbourhood of x (over the aforementioned affine neighbourhood of y) of ring B, such that B is an A-algebra of finite type.
There are many important examples of preschemes locally of finite type over S, that are not of finite type over S, given by solutions of classical universal problems;
thus it is important to be able to consider the Picard scheme of a curve as a union of infinitely-many connected components (that we must avoid confusing with the connected component of the identity element, i.e. the "Picard variety").
It is thus sometimes useful to place ourselves in the category \mathcal {C} of preschemes locally of finite type over S, in order to study the question of representability of a contravariant functor F.
The main goal of these articles is to develop a general technique that allows us to recognise when such a functor F is representable, and to study the properties of the corresponding S-prescheme X by means of the properties of F.
We note in passing that, in this study, we find non-pathological examples of preschemes over S that are not separated over S, notably as "Picard preschemes" of excellent S-schemes;
we must thus refrain from banishing preschemes that are not schemes from algebraic geometry.
Let X be a prescheme locally of finite type over S, and let
F \colon Y \mapsto \operatorname {Hom} _S(Y,X)
be the associated contravariant functor.
We can consider the restriction F_0 of F to the subcategory \mathcal {C}_0 of \mathcal {C} consisting of preschemes Y over S that are Artinian and finite over S:
if S= \operatorname {Spec} ( \Lambda ), then \mathcal {C}_0 is the category dual to the category of Artinian \Gamma-algebras considered in .
If Y= \operatorname {Spec} (A), where A is a local Artinian ring, then Y consists of a single point y living above a closed point s of S, and an S-homomorphism from Y to X (i.e. an element of F(Y)) is defined by the data of a point x \in X over s, along with an \mathscr { O } _s-homomorphism from \mathscr { O } _x to A.
If there exists such a homomorphism, then x is necessarily a closed point of X (since its residue field is algebraic over the residue field of s).
This thus shows that the restriction F_0 of F to "Artinian Y-algebras" is pro-representable, and is represented by the topological Y-algebra whose local components are the completions \widehat { \mathscr { O } _x} of the local rings of X at the points x of X that are closed and live above closed points of Y.
This shows that only knowing F_0 gives precise information about the structure of X (that is, the structure of the completions of its local rings at the aforementioned points).
We note that, even in the case where S is the spectrum of an algebraically closed field, it is only thanks to the systematic consideration of "varieties" Y such that \mathscr { O } _Y may admit nilpotent elements (and, in particular, working with the spectra of local Artinian rings) that we can arrive at the "good formulation" of classical universal problems, and understand the "infinitesimal" aspect.
If we start with a given functor F, and we want to know whether or not it is representable, then studying the functor F_0 (using and ) will give quasi-complete hints;
either, as is often the case (by simply testing, for example, the nature of the sets F(I_K, \xi ) and their functorial behaviour, cf. ), F_0 is already not pro-representable (which explains the failure of attempts made up until now to define varieties of modules in a reasonably natural way for the classification of vector bundles of rank >1);
or we might be able to show that F_0 is indeed representable, but that that vector spaces F(I_K, \xi ) are not of finite dimension, in which case we must be content with the "formal" solution;
or it could be the case that F_0 is indeed representable by a product of complete Noetherian local rings, which gives very strong assumptions for F itself to be representable, and, combined with the analogous properties (but of a more global nature) that we will later develop, will in all likelihood suffice to imply that it is indeed so.
Finally, we come across interesting geometric problems (see and below) where we have only the functor F_0 (not coming from any "global" functor F), and where we will consider ourselves content if we can associate to it a "formal scheme of modules".
To finish these generalities, we note how the theory of schemes explains some apparent anomalies, such as the Igusa surface V whose "Picard variety" P consists of a single point, and for which, however, \operatorname {H} ^1(V, \mathscr { O } _V) \neq0;
in this case, P is a non-trivial "purely infinitesimal" group, i.e. defined by a local algebra \mathscr { O } of finite rank over the base field k and endowed with a diagonal map corresponding to the multiplicative structure of P;
if \mathfrak {m} is the maximal ideal of \mathscr { O }, then the dual of \mathfrak {m}/ \mathfrak {m}^2 is canonically isomorphic to \operatorname {H} ^1(V, \mathscr { O } _V) (cf. below).
It is only when the Picard group is an algebraic group in the classical sense (i.e. simple over the base field k) that the dimension of \operatorname {H} ^1(V, \mathscr { O } _V) (which is always equal to that of \mathfrak {m}/ \mathfrak {m}^2) is equal to that of the Picard group.
2468fga3.ii-c.2fga3.ii-c.2.xmlThe schemes \underline { \operatorname {Hom}}_S(X,Y), \prod _{X/S}Z, \underline { \operatorname {Aut}}(X), etc.C.2fga3.ii-c
Let X and Y be preschemes over S;
for every prescheme T over S, let X_T=X \times _S T and Y_T=Y \times _S T, and consider the set
F(T) = \operatorname {Hom} _T(X_T,Y_T) = \operatorname {Hom} _S(X_T,Y) = \operatorname {Hom} _S(X \times _S T,Y)
as a contravariant functor in T.
If it is representable, then we denote by \underline { \operatorname {Hom} } _S(X,Y) the prescheme over S that represents it, and we then have a functorial isomorphism
\operatorname {Hom} _S(T, \underline { \operatorname {Hom} } _S(X,Y)) \xrightarrow { \sim } \operatorname {Hom} _S(T \times _S X,Y).
There are variants of this universal problem, the solutions to which can be summarised as follows: a prescheme of S-automorphisms of an S-prescheme X (which will be a prescheme in groups), a prescheme of S-homomorphisms from an S-prescheme in groups to another (which will be a prescheme in commutative groups if the latter scheme in groups is commutative), etc.
We can also generalise the definition of \underline { \operatorname {Hom} } _S(X,Y) by considering a prescheme Z over the prescheme X over S, and the functor
F(T) = \operatorname {Hom} _{X_T}(X_T,Z_T)
(the set of "sections" of Z_T over X_T);
if this functor is representable, then the S-prescheme that represents it will be denoted by \Pi _{X/S}Z, and we will thus have, by definition, a functorial isomorphism
\operatorname {Hom} _S(T, \Pi _{X/S}Z) = \operatorname {Hom} _{X_T}(X_T,Z_T).
Setting Z=Y \times _S X, we recover \underline { \operatorname {Hom} } _S(X,Y).
From these definitions follows a formula for the new preschemes thus introduced that is as trivial as it is useful, that we will not give here (given that it holds in every category where products and fibred products exist).
More serious is the question of existence of schemes of the type \underline { \operatorname {Hom} } _S(X,Y).
We note first of all that, for fixed X, \underline { \operatorname {Hom} } _S(X,Y) (resp. \Pi _{X/S}Z) can only exist for all Y over S (resp. for all Z over X) if X is flat over S.
Furthermore, we can convince ourselves that it is not reasonable to expect the existence of a solution, for general enough Y, except in the case where X is further proper over S.
It seems, however, that these conditions are sufficient for the existence of \underline { \operatorname {Hom} } _S(X,Y) and \Pi _{X/S}Z, with the condition that, if necessary, we make some sort of "quasi-projective" hypothesis on Y/S (resp. Z/X);
this is what we can verify anyway in numerous cases (for example, when Y is affine over S, or, by direct elementary constructions, when X is finite over S).
Then and give:
1702Propositionfga3.ii-c.2-proposition-2.1fga3.ii-c.2-proposition-2.1.xml2.1fga3.ii-c.2
Let \Lambda be a Noetherian ring, and X and Y arbitrary preschemes over \Lambda.
Consider the functor
F(A) = \operatorname {Hom} _A(X_A,Y_A)
on the category \mathcal {C}_0 of Artinian \Lambda-algebras.
If X is flat over \Lambda, then this functor is pro-representable.
Furthermore, we can show that, for all A \in \mathcal {C}_0 and all \xi \in F(A), we have a canonical isomorphism
F(I_A, \xi ) = \operatorname {H} ^1 \Big (X_A, \underline { \operatorname {Hom} } _{ \mathscr { O } _{X_A}} \big ( \xi ^*( \Omega _{Y_A/A}^1), \mathscr { O } _{X_A} \big ) \Big )
where \Omega _{Y_A/A}^1 is the sheaf of Kähler 1-differentials of Y_A with respect to A.
Taking A to be a field, we find, using §A, Proposition 5.1 and the finiteness theorem from , the following corollary:
1703Corollaryfga3.ii-c.2-proposition-2.1-corollaryfga3.ii-c.2-proposition-2.1-corollary.xmlfga3.ii-c.2
Suppose that X is flat and proper over S, and that Y is of finite type over S.
Then F is pro-representable, and the local components of the corresponding topological \Lambda-algebra are Noetherian rings.
1704Remarksfga3.ii-c.2-remarksfga3.ii-c.2-remarks.xmlfga3.ii-c.2
The problems considered in this section, and many others, having been generally studied, in the framework of classical algebraic geometry, via the "Chow coordinates" of cycles in projective space, allow us to consider these cycles as points of suitable projective varieties.
This procedure, and, more generally, the use of Chow coordinates, seems irredeemably insufficient from the point of view of schemes, since it destroys the nilpotent elements in the parameterised varieties, and, in particular, do not lend themselves to a satisfying study of infinitesimal variations of cycles (without taking its non-intrinsic nature, linked to the projective space, into account).
The language of Chow coordinates has sadly been the only one used by many algebraic geometers for the study of families of varieties or families of cycles, which seems to have been a serious obstacle to the understanding of these notions, despite its certain technical interest (probably temporary).
If we wish to obtain the analogue of Chow varieties in the theory of schemes, we are led to the following universal problem:
let X be a prescheme over S, and, for every prescheme T over S, consider the set F(T) of closed sub-preschemes of X_T=X \times _S T that are flat over T; we want to represent this functor in T via some prescheme over S.
More generally, we can start with a quasi-coherent sheaf \mathscr { G } on X, and take F(T) to be the set of quotient sheaves of \mathscr { G } _T that are flat over T.
It seems that there exists a solution to this problem, with a scheme C that is locally of finite type over S, if X is proper over S, if S is locally Noetherian, and if F is furthermore coherent.
In any case, supposing only that S is locally Noetherian, the restriction of F to "Artinian S-algebras" is pro-representable, and, if, furthermore, X is proper over S, and F is coherent, then the local components of the corresponding topological ring \mathscr { O } are Noetherian.
Of course, even after having proven the existence of the "Chow scheme" of X over S, it remains to find a decomposition of it into disjoint open subsets C_i (corresponding to fixing continuous invariants, such as degree and dimension of the cycles that we vary) over S, as well as to make precise the relations between this scheme with the classical Chow varieties, and to make precise when a C_i is projective (or at least quasi-projective) over S.
1705Remarkfga3.ii-c.2-remarkfga3.ii-c.2-remark.xmlfga3.ii-c.2[Comp.]
The problems described here are completely resolved in the projective case by "Hilbert schemes" (cf. ).
Examples by Nagata and Hironaka show, however, that the functors described are not necessarily representable if we do not make the projective hypothesis, even if we restrict to the classification of subvarieties of dimension 0 of a complete non-singular variety of dimension 3;
this is linked to the fact that the symmetric square of such a variety does not necessarily exist.
2469fga3.ii-c.3fga3.ii-c.3.xmlPicard schemesC.3fga3.ii-c[Comp.]
For a more complete study, see .
Let f \colon X \to S be an S-prescheme, and consider the multiplicative sheaf \mathscr { O } _X^ \times of units of the structure sheaf of X, along with the group
P(X/S) = \operatorname {H} ^0(S, \operatorname {R} ^1f_*( \mathscr { O } _X^ \times )),
called the relative Picard group of X/S.
An element of this group is thus defined by giving an open cover (U_i) of S, along with an invertible sheaf \mathscr { L } _i on each f^{-1}(U_i), such that \mathscr { L } _i|f^{-1}(U_i \cap U_j) is isomorphic to \mathscr { L } _j|f^{-1}(U_i \cap U_j) for all i,j, or, at least locally over U_i \cap U_j (i.e. these two sheaves are "equivalent" in the sense of FGA 3.I, §B.4).
If X/S admits a section, then P(X/S) is exactly the set of classes of invertible sheaves on X/S up to "equivalence" (loc. cit.).
We now set, for all T over S,
F(T) = P(X_T/T)
which gives a covariant functor in T, that we call the Picard functor of X/S;
if this functor is representable, then the prescheme over S that represents it is called the Picard prescheme of X/S, and denoted by \mathscr { P } (X/S).
In this case, we then have an isomorphism of functors:
\operatorname {Hom} _S(T, \mathscr { P } (X/S)) \xrightarrow { \sim } P(X_T/T).
Taking the Picard prescheme is compatible with extension of the base, and, in particular, the Picard preschemes of the fibres of X over S (which are preschemes over the residue fields K(s) of the s \in S) are the fibres of \mathscr { P } (X/S).
Of course, since P(X_T/T)=F(T) is a commutative group, the Picard preschemes are preschemes in groups.
Note as well that the generalised Jacobians of Rosenlicht are exactly the connected components of the identity in the Picard schemes of complete curves (possibly with singularities), which should make most of their properties clear (once their existence has been proven).
1758Remarkfga3.ii-c.3-remarkfga3.ii-c.3-remark.xmlfga3.ii-c.3
The definition adopted here is only reasonable when every point of Y admits an open neighbourhood U over which X admits a section.
In the general case, it is necessary to slightly modify the definition of the Picard functor in order to still obtain an existence theorem.
Here, the plausible existence conditions for a Picard prescheme are the following: X is proper and flat over S; f_*( \mathscr { O } _X)= \mathscr { O } _S; and X locally admits a section over S.
This condition naturally arises in the application of the technique of descent, in eliminating the automorphisms of an invertible sheaf \mathscr { L } on X by endowing them with a marked section over the section s (FGA 3.I, §B.4).
Notably, we find the following:
1759Propositionfga3.ii-c.3-proposition-3.1fga3.ii-c.3-proposition-3.1.xml3.1fga3.ii-c.3
Suppose that X is flat over S= \operatorname {Spec} ( \Lambda ), where \Lambda is Noetherian, and suppose that, for all T of finite type over S, we have {f_T}_*( \mathscr { O } _{X_T})= \mathscr { O } _T (if f is proper and separable and has separable fibres, or if S is the spectrum of a field, then it follows from Künneth that the latter condition is equivalent to f_*( \mathscr { O } _X)= \mathscr { O } _S).
Then the Picard functor of X/S on the category of Artinian \Lambda-algebras is pro-representable.
Furthermore, we then have
F(I_A, \xi ) = \operatorname {H} ^1(X_A, \mathscr { O } _{X_A}),
and, in particular:
1760Corollaryfga3.ii-c.3-proposition-3.1-corollaryfga3.ii-c.3-proposition-3.1-corollary.xmlfga3.ii-c.3
If X is proper over S, then the local components of the topological \Lambda-algebra corresponding to the Picard functor are Noetherian.
1761Remarksfga3.ii-c.3-remarks-ifga3.ii-c.3-remarks-i.xmlfga3.ii-c.3
We can generalise the definitions and results from this section to the classification of principal bundles on X, with structure group G being a scheme in groups over S that is affine and flat over S, and also commutative.
In the case where G would not be commutative, and thus where the adjoint bundle in groups of a principal bundle (whose sections of the automorphisms of the principal bundle) would no longer be trivial, no longer holds true as it is stated.
We can, however, modify the universal problem in such a way that we again obtain a solution (at least, for now, in formal geometry).
The golden rule to remember, in the context of the current section and in the following, and every time we are looking for "schemes of modules" for classes of objects that are only defined up to isomorphism, is always the following: eliminate the possible automorphisms of the objects that we want to classify, by introducing, if necessary, auxiliary structures (points or elements of marked sections, fixing differential forms, etc.) that we take to be insignificant enough that we do not substantially modify the initial problem.1762Remarksfga3.ii-c.3-remarks-iifga3.ii-c.3-remarks-ii.xmlfga3.ii-c.3[Comp.]
I have recently shown that the formal scheme of modules for an abelian variety over a field is indeed simple over the Witt ring, or, in other words, that every abelian scheme X over a local Artinian ring that is the quotient of another such scheme Y comes, by reduction, from an abelian scheme over Y.
The proof simply uses the variance properties of the obstruction class of the covering, introduced in FGA II, §6.
Recall also that the schemes of modules for curves of genus g or for polarised abelian schemes have been constructed by Mumford (cf. Séminaire Mumford–Tate, Harvard University (1961–62)).
2470fga3.ii-c.4fga3.ii-c.4.xmlFormal modules of a varietyC.4fga3.ii-c
Let \Lambda be a local Noetherian ring of residue field k (more often than not, \Lambda will be equal to k, or to a Cohen p-ring), and let X_0 be a prescheme over k.
For every local Artinian \Lambda-algebra A, consider the set F(A) of isomorphism classes of A-preschemes X that are flat over A, endowed with an isomorphism
1709Equationfga3.ii-c.4-equation-starfga3.ii-c.4-equation-star.xml*fga3.ii-c.4 X \otimes _A k(A) \xleftarrow { \sim } X_0 \otimes _k k(A) \tag{*}
where k(A) is the residue field of A;
of course, the isomorphisms between such flat A-preschemes should respect the above isomorphism given in the structure.
If A is a (not necessary local) Artinian \Lambda-algebra, with local components A_i, then we take F(A) to be the product of the F(A_i).
Then F becomes a multiplicative functor in A, and we call it the functor of modules for X_0 (and \Lambda).
If this functor is representable, then it has a corresponding local topological \Lambda-algebra O, of residue field k, and the formal spectrum of O is called the formal scheme of modules for X_0 (and \Lambda) (cf. for some details on this).
Here, if we wish to apply the technique of descent, the "finite" automorphisms of X_0 are inoffensive, since they have no influence on the existence of automorphisms (in the precise sense above) of A-preschemes X;
the necessary and sufficient condition, if A is not simply a field, for X to not have any non-trivial A-automorphisms is that
\operatorname {H} ^0(X_0, \mathfrak {G}_{X_0/k}) = 0
where \mathfrak {G}_{X_0/k} denotes the sheaf of k-derivations (i.e. the tangent sheaf) of X_0.
We can easily show (at least, if X_0 is simple over k) that
F(I_A, \xi ) = \operatorname {H} ^1(X_A, \mathfrak {G}_{X_A/A}).
We thus conclude, as per usual:
1710Propositionfga3.ii-c.4-proposition-4.1fga3.ii-c.4-proposition-4.1.xml4.1fga3.ii-c.4
Suppose that \operatorname {H} ^0(X_0, \mathfrak {G}_{X_0/k})=0.
Then the formal scheme of modules for X_0 exists.
If, furthermore, X_0 is proper over k, then the formal scheme of modules is Noetherian.
1711Remarksfga3.ii-c.4-remarksfga3.ii-c.4-remarks.xmlfga3.ii-c.4
If X_0 is not assumed to be simple over k, then F(I_A, \xi ) can be identified with a sub-A-module of
\operatorname {Ext} _{ \mathscr { O } _{P_A}}^1(P_A; \mathscr { I } _{X_A}, \mathscr { O } _{X_A})
where we set P_A=X_A \times _A X_A, where \mathscr { O } _{X_A} is considered as a coherent sheaf on P_A via the diagonal morphism X_A \to P_A, and where \mathscr { I } _{X_A} denotes the coherent sheaf of ideals on P_A defined by the diagonal morphism.
More precisely, an easy globalisation of Hochschild theory shows that the \operatorname {Ext} ^1 above can be identified with the set of classes, up to isomorphism, of sheaves of I_A-algebras \mathscr { O } that are flat over X_A, and endowed with an augmentation isomorphism \mathscr { O } \otimes _{I_A}A \to \mathscr { O } _{X_A} (recall that we set I_A=At/(t^2)).
The submodule F(I_A, \xi ) is that which corresponds to the sheaves of commutative algebras.
The simplicity hypotheses are thus not essential in the theory of modules, as implies.
Recall (loc. cit.) that, in particular, every simple and proper algebraic curve X_0 over k admits a formal scheme of modules that is simple over \Lambda, and of relative dimension equal to 3g-3 if the genus g is \geqslant2, and to g if g=0,1.
These two latter cases no longer figure directly in .
We can, however, recover them in the case of elliptic curves (g=1) thanks to the remarks that will follow.
We can, of course, vary ad libitum by considering systems of schemes over k endowed with various structures.
Suppose, for example, that X_0 is an abelian scheme over k, with a marked origin (i.e. X_0 is considered as a scheme in groups over k), and let F(A) be the set of isomorphism classes of abelian schemes over A (i.e. of schemes in groups that are proper and simple over A) endowed with an isomorphism as in of abelian schemes.
We can show that imposing a multiplicative structure (or even only a "unit section") eliminates the infinitesimal automorphisms, and that there thus exists a formal scheme of modules that corresponds to a complete local Noetherian ring O.
We can also show that, if X is a proper and simple scheme with "absolutely connected" fibres over a locally Noetherian prescheme S, then every multiplicative structure on X that admits a unit section is necessarily associative and commutative (provided that it is associative and commutative on one fibre, and provided that S is connected), and is furthermore uniquely determined by the knowledge of the unit section.
Further, supposing that S is the spectrum of a local Artinian ring A of residue field k, that X is proper over A and endowed with a section s, and finally that X \otimes _A k is endowed with the structure of an abelian scheme over k, admitting the point of X \otimes _A k corresponding to s as the zero element, an easy calculation of obstructions, combined with an argument due to Serre, allows us to prove that there exists on X a multiplicative structure admitting the section s as the unit section.
(From here, using the "existence theorem" of to pass to the case where A is complete local Noetherian, and then the technique of descent from for the general case, we can prove the analogous claim for all locally Noetherian connected S).
This proves that the functor F(A) considered here is isomorphic to the analogous functor defined at the start of this section by abstracting the multiplicative structure on X_0.
It then follows that, in particular, if \mathfrak {m} is the maximal ideal of O, then \mathfrak {m}/ \mathfrak {m}^2 is canonically isomorphic to the dual of \operatorname {H} ^1(X_0, \mathfrak {G}_{X_0/k}), and is thus of dimension n^2, where n= \dim X_0.
It would be very interesting to determine if O is indeed simple over \Lambda, i.e. isomorphic to an algebra of formal series in n^2 variables over \Lambda.
Now §A, Proposition 5.2 allows us to give an equivalent formulation of this problem as an existence problem of abelian schemes that are reducible along a given abelian scheme.
In any case, we see, by a transcendental way, that the answer is affirmative if k is of characteristic 0.
In characteristic p \neq0, it evidently suffices to restrict to the case where \Lambda is the ring of Witt vectors constructed over an algebraically closed field k.
This could be the moment for the "Greenberg functor" to prove its worth...
2471fga3.ii-c.5fga3.ii-c.5.xmlExtension of coveringsC.5fga3.ii-c
Let \mathfrak {X} be a formal Noetherian prescheme [], U an open subset of \mathfrak {X} defined locally by the "non-vanishing" of a section of \mathscr { O } _ \mathfrak {X} that is not a zero divisor (and thus large enough that every section of \mathscr { O } _ \mathfrak {X} over an open subset V that is zero on U \cap V is zero).
([Comp.] It is also necessary to assume that the section defining U is not a zero divisor not only on \mathfrak {X}, but also on every X_n.)
Let \mathfrak {J} be an "ideal of definition" for \mathfrak {X}, and let X_n=( \mathfrak {X}, \mathscr { O } _ \mathfrak {X}/ \mathfrak {J}^{n+1}), which is thus a ordinary Noetherian prescheme.
Then, if \mathfrak {X}' and \mathfrak {X}'' are flat coverings of \mathfrak {X} (i.e. preschemes over \mathfrak {X} defined by sheaves of algebras that are coherent and locally free as sheaves of modules) that are unramified over U, the evident map
\operatorname {Hom} _ \mathfrak {X}( \mathfrak {X}', \mathfrak {X}'') \to \operatorname {Hom} _{X_0}(X'_0,X''_0)
is injective;
in particular, an automorphism of \mathfrak {X}' that induces the identity on X'_0 is the identity.
This allows us to apply the technique of descent to the situation.
We start, in particular, with a flat covering X'_0 of X_0, unramified over U_0, and let G( \mathfrak {X}) be the set of classes, up to isomorphism (inducing the identity on X'_0), of flat coverings \mathfrak {X}' of \mathfrak {X} that induce X'_0 on X_0 (and that are thus necessarily unramified over U).
We similarly define G(V) for every open subset V of \mathfrak {X}, and, more generally, G( \mathfrak {Y}) for every formal prescheme \mathfrak {Y} over \mathfrak {X}.
With this, the results of and imply, first of all, the following results:
If V varies amongst open subset of \mathfrak {X}, then the G(V) form a sheaf on \mathfrak {X}, say \mathscr { G } _ \mathfrak {X}= \mathscr { G }.
The restriction of this sheaf to U is the constant sheaf whose fibres consist of a single element.
More generally, describing the fibres of \mathscr { G } _ \mathfrak {X} is a question of complete local rings, in a precise way:
For all x \in \mathfrak {X}, we have
\mathscr { G } _x = G( \operatorname {Spec} ( \mathscr { O } _{ \mathfrak {X},x})) \subset G( \operatorname {Spec} ( \widehat { \mathscr { O } }_{ \mathfrak {X},x}))
(i.e. isomorphism classes of finite free algebras B over \widehat { \mathscr { O } }_{ \mathfrak {X},x} endowed with an isomorphism from B \otimes _{ \widehat { \mathscr { O } }_{ \mathfrak {X},x}} \mathscr { O } _{X_0,x} to ( \widehat { \mathscr { O } }'_0)_x, where \mathscr { O } '_0 is the sheaf of algebras on X_0 that defines X'_0).
We have a canonical isomorphism \mathscr { G } _ \mathfrak {X}= \varprojlim \mathscr { G } _{ \mathfrak {X}_n}; in other words, for every open subset V of \mathfrak {X}, we have G(V)= \varprojlim G(V_n).
Suppose that \mathfrak {X} comes from an ordinary proper scheme X over a complete local Noetherian ring \Lambda that has ideal of definition \mathfrak {m} by taking the \mathscr { J }-adic completion of \mathscr { O } _X, where \mathscr { J } = \mathfrak {m} \cdot \mathscr { O } _X.
Then G( \mathfrak {X}) is canonically isomorphic to the set of classes of flat coverings of the ordinary scheme X that are "reducible along X'_0".
Figuratively speaking, we can say that (a) and (b) establish the fundamental relations between the local and global aspects of the problem; (c) gives the relations between the "finite" and "infinitesimal" aspects; and finally (d) remembers (under precise conditions) the identity between the "formal" and "algebraic" aspects.
Now suppose that \mathfrak {X} is defined by a local complete Noetherian ring \Lambda, with \mathscr { J } = \mathfrak {m} \cdot \mathscr { O } _X (and so X_0 is a prescheme over { \Lambda }/ \mathfrak {m}).
For every algebra A that is finite over \Lambda, we set
F(A) = G( \mathfrak {X} \times _ \Lambda A).
This is a covariant functor in A, with values in the category of sets, and, by (c), this functor is completely determined by how it acts on Artinian algebras A;
it is equivalent to say either that this functor is pro-representable, i.e. of the form
F(A) = \operatorname {Hom} _{ \text {top. } \Lambda \text {-algebras}}( \mathscr { O } ,A)
where \mathscr { O } is a topological \Lambda algebra of the type described in , or that this is true when we restrict to only Artinian algebras A.
The combination of and then effectively implies:
1722Propositionfga3.ii-c.5-proposition-5.1fga3.ii-c.5-proposition-5.1.xml5.1fga3.ii-c.5
The above functor is pro-representable.
Of course, by (a), if U= \mathfrak {X}, then G( \mathfrak {Y}) consists of a single element for all \mathfrak {Y} over \mathfrak {X}, and the functor F is then not very interesting (we will have \mathscr { O } = \Lambda).
It seems that, in practically every other case, the topological local ring \mathscr { O } is not Noetherian.
Its existence, however, shows, in a striking manner, the "continuous" nature of the set G( \mathfrak {X}) of solutions (corresponding intuitively to the fact that there is a "continuous" choice in the way in which the ramification spreads when we take an extension of X'_0).
We will compare this result with the point of view of J.-P. Serre [@Ser1958] via local class field theory, drawing attention as well to the continuous character of the topological Galois group of the maximal abelian extension of a "geometric" local field, with the dual group (in the sense of Pontrjagin) appearing as an inductive limit of algebraic (or at least quasi-algebraic) groups;
here as well, the classification of extensions is given by infinite-dimensional "varieties".
We can also take, in the above, \mathfrak {X} to be the formal spectrum of a complete local ring (of which \Lambda will be, for example, a Cohen subring), and we might hope that the results of this section can be used in the study of extensions of a local complete ring of dimension >1.
Just as much in the local case as in the global case, they might allow us to formulate precise relations between the phenomena of higher ramification and phenomena in characteristic 0 (approachable via a transcendental way).
In any case, it is the preliminary analysis of that allows us to extend the methods described in for the study of the fundamental group to the "tamely ramified" case, and to resolve, by a transcendental way, the "problem of three points".
To finish, we note that the situation simplifies if X_0 is of dimension 1;
then, by (a) and (b), G( \mathfrak {X}) can be identified with \prod _i G( \operatorname {Spec} ( \mathscr { O } _{ \mathfrak {X},x_i})), where the x_i are the points of X_0 \setminus U:
we can take arbitrary "local" extensions at the ramification points.
Further, if X_0 is normal, then we note that the formal scheme of modules guaranteed by is simple over \operatorname {Spec} ( \Lambda ).
3401fga3.iii-2fga3.iii-2.xmlQuotient preschemes › Example: finite preschemes over S2fga3.iii
Let \mathcal {C} be the category of finite preschemes over S, which is assumed to be locally Noetherian.
Then \mathcal {C} is equivalent to the opposite category of the category of coherent sheaves of commutative algebras on S, or, if S is affine of ring A, then it is equivalent to the opposite category of the category of finite A-algebras over A (i.e. those that are modules of finite type over A).
We thus immediately conclude that, in \mathcal {C}, finite projective limits and finite inductive limits exist.
This is well known (without any finiteness hypotheses) for the former;
the fibre product of preschemes X and Y over S corresponds to the tensor product B \otimes _A C of corresponding algebras, and the kernel of two morphisms X \rightrightarrows Y, defined by two A-algebra homomorphisms u,v \colon C \rightrightarrows B, corresponds to the quotient of B by the ideal generated by the u(v)-v(c), etc.
For finite inductive limits, it suffices to consider, on one hand, finite sums, which correspond to the ordinary product of A-algebras, and, on the other hand, cokernels of pairs of morphisms X \rightrightarrows Y, which correspond (as we can immediately see) to the sub-ring of C given by elements where the homomorphisms u,v \colon C \rightrightarrows B agree (with this sub-ring being finite over A thanks to the Noetherian hypothesis).
We also note that we can show, using the Noetherian hypothesis, that finite inductive limits, and, in particular, quotients, thus constructed in the category \mathcal {C} of finite preschemes over S are, in fact, quotients in the category of all preschemes.
As we mentioned in , there are non-effective epimorphisms in \mathcal {C} (or even non-strict, which is the same, since fibre products exist).
I do not know if equivalence relations are still effective if we have no flatness hypothesis.
I have only obtained, in this direction, very partial, positive, results, that are vital for the proof of the fundamental theorem of the formal theory of modules (cf. FGA 3.II, §B, Theorem 1).
We note that it is easy, in the given problem, to reduce to the case where S is the spectrum of a local Artinian ring, with an algebraically closed residue field.
But even if A is a field, the answer is not known.
We can also consider the case of a prescheme X over S that is no longer assumed to be finite over S, but by considering an equivalence relation R on X such that p_1 \colon R \to X is a finite morphism.
We then say that R is a finite equivalence relation.
Supposing, for simplicity, that S and X are affine (which implies that R is affine, so that the situation is reduced to one of pure commutative algebra), we do not know, even in this case, if there exists a quotient X/R=Y, and if the canonical morphism X \to Y is finite.
(The most simple case is that where we suppose that S is the spectrum of a field k, and where X is the spectrum of k[t], i.e. the affine line).
Of course, if the two problems above turn out to be true, then we can conclude that, in the situation described, R is effective.
Note that the problem of existence of a quotient Y and of the finiteness of f \colon X \to Y are stated in exactly the same terms if, instead of an equivalence graph in X, we only have an equivalence pregraph in X, in the sense of .
The question of passing to the quotient by a more or less arbitrary finite equivalence relation arises in the construction of preschemes by "gluing" given preschemes X_i along certain closed sub-preschemes;
the gluing law is expressed precisely by a finite equivalence relation on the prescheme X given by the sum of the X_i.
We also expect that the solutions of the problems stated here, as well as of their many variations, will be a preliminary condition for the clarification of a general technique for non-projective constructions, in the direction introduced in .
The only general positive fact known to the author is the following:
1715Propositionfga3.iii-2-proposition-2.1fga3.iii-2-proposition-2.1.xml2.1fga3.iii-2
Let S be a locally Noetherian prescheme, s a point of S, and \Omega an algebraically closed extension of k(s).
Consider the corresponding "fibre functor" F, that associates, to any S-scheme X that is finite over S, the set of points of X/S with values in \Omega.
This functor (which is trivially left exact) is _right exact_, i.e. it commutes with finite inductive limits, and, in particular, with the cokernel of pairs of morphisms.
By using this result for all the "geometric points" of S, we thus deduce that the "quotient" category \mathcal {C}' of \mathcal {C}, given by arguing "modulo surjective radicial morphisms" (i.e. by formally adjoining inverses for such morphisms), is a "geometric" category, i.e. it satisfies the same "finite nature" properties as the category of sets.
In particular, every equivalence relation is effective.
This implies that, if R is an equivalence relation on X, where X is finite over S, then the canonical morphism R \to X \times _Y X (where Y=X/R) is radicial and surjective (and, in fact, a surjective closed immersion, since it is a monomorphism).
3402FGAfga3.ifga3.i.xmlGeneralities, and descent by faithfully flat morphisms3.I
A. Grothendieck.
"Technique de descente et théorèmes d'existence en géométrie algébrique, I: Généralités. Descente par morphismes fidèlement plats".
Séminaire Bourbaki 12 (1959–60), Talk no. 190.
(Numdam)
[Comp.]
For various details concerning the theory of descent, see also [Gro1960b, VI, VII, and VIII].
From a technical point of view, the current article, and those that will follow, can be considered as variations on Hilbert's celebrated "Theorem 90".
The introduction of the method of descent in algebraic geometry seems to be due to A. Weil, under the name of "descent of the base field".
Weil considered only the case of separable finite field extensions.
The case of radicial extensions of height 1 was then studied by P. Cartier.
Lacking the language of schemes, and, more particularly, lacking nilpotent elements in the rings that were under consideration, the essential identity between these two cases could not have been formulated by Cartier.
Currently, it seems that the general technique of descent that will be explained (combined with, when necessary, the fundamental theorems of "formal geometry", cf. ) is at the base of the majority of existence theorems in algebraic geometry.
([Trans.] [Comp.] It now seems excessive to say that the technique of descent is "at the base of the majority of existence theorems in algebraic geometry". This is true to a large extent for the non-projective techniques that are the object of study of the first two talks of this current series (i.e. "Techniques of descent and existence theorems in algebraic geometry"), but not for the projective techniques (talks IV, V, and VI).)
It is worth noting as well that this aforementioned technique of descent can certainly be transported to "analytic geometry", and we can hope that, in the not-too-distant future, specialists will know how to prove the "analytic" analogues of the existence theorems in formal geometry that will be given in talk II.
In any case, the recent work of Kodaira–Spencer, whose methods seem unfit for defining and studying "varieties of modules" in the neighbourhood of their singular points, shows reasonably clearly the necessity of methods that are closer to the theory of schemes (which should naturally complement transcendental techniques).
In the present talk (namely talk I) we will discuss the most elementary case of descent (the one indicated in the title).
The applications of , , and below (in ) are, however, already vast in number.
We will restrict ourselves to giving only some of them as examples, without aiming for the maximum generality possible.
We will freely use the language of schemes, for which we refer to the already cited article, as well as [GR1958].
We make clear to point out, however, that the preschemes considered in this current article are not necessarily Noetherian, and that the morphisms are not necessarily of finite type.
So, if A is a local Noetherian ring, with completion \overline {A}, then we will need to consider the non-Noetherian ring \overline { \overline {A}} \otimes _A \overline {A}, as well as the morphisms of affine schemes that correspond to the inclusions between the rings in question.
587fga3.i-afga3.i-a.xmlPreliminaries on categoriesAfga3.i552fga3.i-a.1fga3.i-a.1.xmlFibred categories, descent data, \mathcal {F}-descent morphismsA.1fga3.i-a535fga3.i-a.1.afga3.i-a.1.a.xmlA.1.afga3.i-a.1532Definitionfga3.i-a.1-definition-1.1fga3.i-a.1-definition-1.1.xml1.1fga3.i-a.1.a
A fibred category \mathcal {F} with base \mathcal {C} (or over \mathcal {C}) consists of
a category \mathcal {C}
for every X \in \mathcal {C}, a category \mathcal {F}_X
for every \mathcal {C}-morphism f \colon X \to Y, a functor f^* \colon \mathcal {F}_Y \to \mathcal {F}_X, which we also write as
f^*( \xi ) = \xi \times _Y X
for \xi \in \mathcal {F}_Y (with X being thought of as an "object of \mathcal {C} over Y", i.e. as being endowed with the morphism f)
for any two composible morphisms X \xrightarrow {f}Y \xrightarrow {g}Z, an isomorphism of functors
c_{f,g} \colon (gf)^* \to f^*g^*
with the above data being subject to the conditions that
\operatorname {id} ^*= \operatorname {id}
c_{f,g} is the identity isomorphism if f or g is an identity isomorphism
for any three composible morphisms X \xrightarrow {f}Y \xrightarrow {g}Z \xrightarrow {h}T, the following diagram, given by using the isomorphisms of the form c_{u,v}, commutes:
\begin {CD} (h(gf))^* @= ((hg)f)^* \\ @VVV @VVV \\ (gf)^*h^* @. f^*(hg)^* \\ @VVV @VVV \\ (f^*g^*)h^* @= f^*(g^*h^*) \end {CD}
533Examplefga3.i-a.1-example-1fga3.i-a.1-example-1.xml1fga3.i-a.1.a
Let \mathcal {C} be a category where all fibre products exist.
We then define a fibred category \mathcal {F} with base \mathcal {C} by setting \mathcal {F}_X to be the category of objects of \mathcal {C} over X, and the functor f^* \colon \mathcal {F}_Y \to \mathcal {F}_X corresponding to a morphism f \colon X \to Y being defined by the fibre product Z \mapsto Z \times _Y X.
534Examplefga3.i-a.1-example-2fga3.i-a.1-example-2.xml2fga3.i-a.1.a
Let \mathcal {C} be the category of preschemes, and, for X \in \mathcal {C}, let \mathcal {F}_X be the category of quasi-coherent sheaves of modules on X.
If f \colon X \to Y is a morphism of preschemes, then f^* \colon \mathcal {F}_Y \to \mathcal {F}_X is the inverse image of sheaves of modules functor.
We thus obtain a category fibred over \mathcal {C}.
541fga3.i-a.1.bfga3.i-a.1.b.xmlA.1.bfga3.i-a.1536Definitionfga3.i-a.1-definition-1.2fga3.i-a.1-definition-1.2.xml1.2fga3.i-a.1.b
A diagram of maps of sets
E \xrightarrow {u} E' \overset { v_1 }{ \underset { v_2 }{ \rightrightarrows }} E''
is said to be exact if u is a bijection from E to the subset of E' consisting of the x' \in E' such that v_1(x')=v_2(x').
538Definitionfga3.i-a.1-definition-1.3fga3.i-a.1-definition-1.3.xml1.3fga3.i-a.1.b
Let \mathcal {F} be a fibred category with base \mathcal {C}, and consider a diagram of morphisms in \mathcal {C}
S \xleftarrow { \alpha } S' \overset { \beta _1 }{ \underset { \beta _2 }{ \leftleftarrows }} S''
such that \alpha \beta _1= \alpha \beta _2;
this diagram is said to be \mathcal {F}-exact if, for every pair ( \xi , \eta ) of elements of \mathcal {F}_S, the diagram of sets
537Equationfga3.i-a.1-definition-1.3-equationfga3.i-a.1-definition-1.3-equation.xml+fga3.i-a.1-definition-1.3 \operatorname {Hom} ( \xi , \eta ) \xrightarrow { \alpha ^*} \operatorname {Hom} ( \alpha ^*( \xi ), \alpha ^*( \eta )) \overset { \beta _1^* }{ \underset { \beta _2^* }{ \rightrightarrows }} \operatorname {Hom} ( \gamma ^*( \xi ), \gamma ^*( \eta )) \tag{+}
(where \gamma = \alpha \beta _1= \alpha \beta _2) is exact.
In this diagram above, for simplicity, we have identified \beta _i^* \alpha ^* with ( \alpha \beta _i)^*= \gamma ^*, using c_{ \beta _i, \alpha }.
539Definitionfga3.i-a.1-definition-1.4fga3.i-a.1-definition-1.4.xml1.4fga3.i-a.1.b
Let \mathcal {F} be a fibred category with base \mathcal {C}, and consider morphisms \beta _1, \beta _2 \colon S'' \to S' in \mathcal {C}.
Let \xi ' \in \mathcal {F}_{S'}.
We define a gluing data on \xi ' (with respect to the pair ( \beta _1, \beta _2)) to be an isomorphism from \beta _1^*( \xi ') to \beta _2^*( \xi ').
If \xi ', \eta ' \in \mathcal {F}_{S'} are both endowed with gluing data, then a morphism u \colon \xi ' \to \eta ' in \mathcal {F}_{S'} is said to be compatible with the gluing data if the following diagram commutes:
\begin {CD} \beta _1^*( \xi ') @>>> \beta _2^*( \xi ') \\ @VVV @VVV \\ \beta _1^*( \eta ') @>>> \beta _2^*( \eta '). \end {CD}
With this definition, the objects of \mathcal {F}_{S'} that are endowed with gluing data (with respect to \beta _1 and \beta _2) then form a category.
If \alpha \colon S' \to S is a morphism such that \alpha \beta _1= \alpha \beta _2, then, for every \xi \in \mathcal {F}_{S'}, the object \xi '= \alpha ^*( \xi ) of \mathcal {F}_{S'} is endowed with a canonical gluing data, since
\beta _i^* \alpha ^*( \xi ) \simeq ( \alpha \beta _i)^*( \xi ) = \gamma ^*( \xi ),
where we again set \gamma = \alpha \beta _1= \alpha \beta _2;
furthermore, if u \colon \xi \to \eta is a morphism in \mathcal {F}_s, then
\alpha ^*(u) \colon \alpha ^*( \xi ) \to \alpha ^*( \eta )
is a morphism in \mathcal {F}_{S'} that is compatible with the canonical gluing data.
We thus obtain a canonical functor from the category \mathcal {F}_S to the category of objects of \mathcal {F}_{S'} endowed with gluing data with respect to the pair ( \beta _1, \beta _2).
With this, we can also rephrase by saying that is \mathcal {F}-exact if the above functor is fully faithful, i.e. if the above functor defines an equivalence between the category \mathcal {F}_S and a subcategory of the category of objects of \mathcal {F}_{S'} endowed with gluing data with respect to ( \beta _1, \beta _2).
540Definitionfga3.i-a.1-definition-1.5fga3.i-a.1-definition-1.5.xml1.5fga3.i-a.1.b
We say that a gluing data on \xi ' \in \mathcal {F}_{S'} is effective (with respect to \alpha) if \xi ', endowed with this data, is isomorphic to \alpha ^*( \xi ) for some \xi \in \mathcal {F}_S.
In the case where is \mathcal {F}-exact, the object \xi in is then determined up to unique isomorphism, and the category \mathcal {F}_S is equivalent to the category of objects of \mathcal {F}_{S'} endowed with effective gluing data.
547fga3.i-a.1.cfga3.i-a.1.c.xmlA.1.cfga3.i-a.1
The most important case is that where
S'' = S' \times _S S',
with the \beta _i being the two projections p_1 and p_2 from S' \times _S S' to its two factors (where we now suppose that \mathcal {C} has all fibre products).
We then have two immediate necessary conditions for a gluing data \varphi \colon p_1^*( \xi ') \to p_2^*( \xi ') on some \xi ' \in \mathcal {F}_S to be effective:
\Delta ^*( \varphi ) = \operatorname {id} _ \xi, where \Delta \colon S' \to S' \times _S S' denotes the diagonal morphism, and where we identify \Delta ^* p_i^*( \xi ') with (p_i \Delta )^*( \xi ')= \xi '.
p_{31}^*( \varphi ) = p_{32}^*( \varphi )p_{21}^*( \varphi ), where p_{ij} denotes the canonical projection from S' \times _S S' \times _S S' to the partial product of its ith and jth factors.
545Definitionfga3.i-a.1-definition-1.6fga3.i-a.1-definition-1.6.xml1.6fga3.i-a.1.c
We define descent data on \xi ' \in \mathcal {F}_{S'}, with respect to the morphism \alpha \colon S' \to S, to be a gluing data on \xi ' with respect to the pair (p_1,p_2) of canonical projections S' \times _S S' \to S' that satisfies conditions (i) and (ii) above.
546Definitionfga3.i-a.1-definition-1.7fga3.i-a.1-definition-1.7.xml1.7fga3.i-a.1.c
A morphism \alpha \colon S' \to S is said to be an \mathcal {F}-descent morphism if the diagram of morphisms
S \xleftarrow { \alpha } S' \overset { p_1 }{ \underset { p_2 }{ \leftleftarrows }} S' \times _S S'
is \mathcal {F}-exact ();
we say that \alpha is a strict \mathcal {F}-descent morphism if, further, every descent data () on any object of \mathcal {F}_{S'} is effective.
This latter condition (of strictness) can also be stated in a more evocative way:
"giving an object of \mathcal {F}_S is equivalent to giving an object of \mathcal {F}_{S'} endowed with a descent data".
Note that, if an \mathcal {F}-descent morphism
([Comp.] It is useless to assume here that \alpha is an \mathcal {F}-descent morphism.)
\alpha \colon S' \to S admits a section s \colon S \to S' (i.e. a morphism s such that \alpha s= \operatorname {id} _S), then it is a strict \mathcal {F}-descent morphism:
if \xi ' \in \mathcal {F}_{S'} is endowed with descent data with respect to \alpha, then "it comes from" \xi =s^*( \xi ').
551fga3.i-a.1.dfga3.i-a.1.d.xmlA.1.dfga3.i-a.1
We can present the above notions in a more intuitive manner, by introducing, for an object T of \mathcal {C} over S, the set
\operatorname {Hom} _S(T,S') = S'(T),
where elements will be denoted by t, t', etc.
Given an object \xi ' \in \mathcal {F}_{S'}, there then corresponds, to every t \in S'(T), an object t^*( \xi ') of \mathcal {F}_T, which will also be denoted by \xi '_t.
A gluing data on \xi ' (with respect to (p_1,p_2)) is then defined by the data, for every T over S, and every pair of points t,t' \in S'(T), of an isomorphism
\varphi _{t',t} \colon \xi '_t \to \xi '_{t'}
(satisfying the evident conditions of functoriality in T).
Conditions (i) and (ii) of can then be written as
\varphi _{t,t}= \operatorname {id} _{ \xi '_t}, for all T and all t \in S'(T).
\varphi _{t,t''}= \varphi _{t,t'} \varphi _{t',t''}, for all T and all t,t',t'' \in S'(T).
We can show that (ii bis) implies that \varphi _{t,t}^2= \varphi _{t,t}, by taking t=t'=t'', and thus, since \varphi _{t,t} is an isomorphism by hypothesis, implies (i bis), which is thus a consequence of (ii bis) (and so (i) is also a consequence of (ii)).
But if we no longer suppose a priori that the \varphi _{t,t} are isomorphisms (i.e. that \varphi \colon p_1^*( \xi ') \to p_2^*( \xi ') is an isomorphism), then (ii bis) no longer necessarily implies (i bis);
the combination of (ii bis) and (i bis), however, does imply that the \varphi _{t,t'} are isomorphisms (since we then have \varphi _{t,t'} \varphi _{t',t}= \varphi _{t,t}= \operatorname {id} _{ \xi '_t}).
564fga3.i-a.2fga3.i-a.2.xmlExact diagrams and strict epimorphisms, descent morphisms, and examplesA.2fga3.i-a558fga3.i-a.2.afga3.i-a.2.a.xmlA.2.afga3.i-a.2553Definitionfga3.i-a.2-definition-2.1fga3.i-a.2-definition-2.1.xml2.1fga3.i-a.2.a
Let \mathcal {C} be a category.
A diagram of morphisms
T \xrightarrow { \alpha } T' \overset { \beta _1 }{ \underset { \beta _2 }{ \rightrightarrows }} T''
is said to be exact if, for all Z \in \mathcal {C}, the corresponding diagram of sets
\operatorname {Hom} (Z,T) \to \operatorname {Hom} (Z,T') \rightrightarrows \operatorname {Hom} (Z,T'')
is exact ().
We then say that (T, \alpha ) (or, by an abuse of language, T) is a kernel of the pair ( \beta _1, \beta _2) of morphisms.
This kernel is evidently determined up to unique isomorphism.
If \mathcal {C} is the category of sets, then the above definition is compatible with .
Dually, we define the exactness of a diagram of morphisms in \mathcal {C}
S \xleftarrow { \alpha } S' \overset { \beta _1 }{ \underset { \beta _2 }{ \leftleftarrows }} S''
and then say that (S, \alpha ) is a cokernel of the pair ( \beta _1, \beta _2) of morphisms.
554Definitionfga3.i-a.2-definition-2.2fga3.i-a.2-definition-2.2.xml2.2fga3.i-a.2.a
A morphism \alpha \colon S' \to S is said to be a strict epimorphism if it is an epimorphism and, for every morphism u \colon S' \to Z, the following necessary condition is also sufficient for u to factor as S' \to S \to Z:
for every S'' \in \mathcal {C} and every pair \beta _1, \beta _2 \colon S'' \to S of morphisms such that \alpha \beta _1= \alpha \beta _2, we also have that u \beta _1=u \beta _2.
If the fibre product S' \times _S S' exists, then it is equivalent to say that the diagram
S \xleftarrow { \alpha } S' \overset { p_1 }{ \underset { p_2 }{ \leftleftarrows }} S' \times _S S'
is exact, i.e. that S is a cokernel of the pair (p_1,p_2).
In any case, a cokernel morphism is a strict epimorphism.
Note also that, if a strict epimorphism is also a monomorphism, then it is an isomorphism.
We leave to the reader the task of developing the dual notion of a strict monomorphism.
To make the relation between the ideas of \mathcal {F}-descent morphisms and strict epimorphisms more precise, we introduce the following definitions:
555Definitionfga3.i-a.2-definition-2.3fga3.i-a.2-definition-2.3.xml2.3fga3.i-a.2.a
A morphism \alpha \colon S' \to S is said to be a universal epimorphism (resp. a strict universal epimorphism) if, for every T over S, the fibre product T'=S' \times _S T exists, and the projection T' \to T is an epimorphism (resp. a strict epimorphism).
In very nice categories (such as the category of sets, the category of sets over a topological space, abelian categories, etc.), the four notions of "epijectivity" introduced above all coincide;
they are, however, all distinct in a category such as the category of preschemes, or the category of preschemes over a given non-empty prescheme S, even if we restrict to S-schemes that are finite over S.
556Definitionfga3.i-a.2-definition-2.4fga3.i-a.2-definition-2.4.xml2.4fga3.i-a.2.a
A morphism \alpha \colon S' \to S is said to be a descent morphism (resp. a strict descent morphism) if it is an \mathcal {F}-descent morphism (resp. a strict \mathcal {F}-descent morphism) (cf. ), where \mathcal {F} denotes the fibred category over \mathcal {C} of objects of \mathcal {C} over objects of \mathcal {C} (cf. ).
557Propositionfga3.i-a.2-proposition-2.1fga3.i-a.2-proposition-2.1.xml2.1fga3.i-a.2.a
If \mathcal {C} has all finite products and (finite) fibre products, then there is an identity between descent morphisms in \mathcal {C} and strict universal epimorphisms in \mathcal {C}.
561fga3.i-a.2.bfga3.i-a.2.b.xmlA.2.bfga3.i-a.2560Examplefga3.i-a.2.b
Let \mathcal {C} be the category of preschemes.
Let S \in \mathcal {C}, and let S' and S'' be preschemes that are finite over S, i.e. that correspond to sheaves of algebras \mathscr { A } ' and \mathscr { A } '' over S that are quasi-coherent (as sheaves of modules) and of finite type (i.e. coherent, if S is locally Noetherian).
Let \alpha \colon S' \to S be the structure morphism of S', and let \beta _1 and \beta _2 be S-morphisms from S'' to S', defined by algebra homomorphisms \mathscr { A } ' \to \mathscr { A } '', which we also denote by \beta _1 and \beta _2.
Using the fact that a finite morphism is closed (the first Cohen–Seidenberg theorem), we can easily prove that the diagram in \mathcal {C}
559Equationfga3.i-a.2.b-equationfga3.i-a.2.b-equation.xml+ S \xleftarrow { \alpha } S' \overset { \beta _1 }{ \underset { \beta _2 }{ \leftleftarrows }} S'' \tag{+}
is exact if and only if the diagram of sheaves on S
\mathscr { O } _S = \mathscr { A } \xrightarrow { \alpha } \mathscr { A } ' \overset { \beta _1 }{ \underset { \beta _2 }{ \rightrightarrows }} \mathscr { A } ''
is exact.
In particular, if \alpha \colon S' \to S is a finite morphism corresponding to a sheaf \mathscr { A } ' of algebras on S, then \alpha is a strict epimorphism if and only if the diagram of sheaves
\mathscr { O } _S = \mathscr { A } \to \mathscr { A } ' \overset { p_1 }{ \underset { p_2 }{ \rightrightarrows }} \mathscr { A } ' \otimes _{ \mathscr { A } } \mathscr { A } '
is exact (it is an epimorphism if and only if \mathscr { A } \to \mathscr { A } ' is injective).
If S is affine of ring A, then S' is affine of ring A', with A' finite over A, and so S' \to S is a strict epimorphism if and only if A \to A' is an isomorphism from A to the subring of A' consisting of the x' \in A' such that
1_{A'} \otimes _A x' - x' \otimes _A 1_{A'} = 0
(it is an epimorphism if and only if A \to A' is injective).
As we have already mentioned, even if S is the scheme of a local Artinian ring, then a finite morphism S' \to S that is an epimorphism is not necessarily a strict epimorphism.
However, we can prove that, if S is a Noetherian prescheme, then every finite morphism S' \to S that is an epimorphism is the composition of a finite sequence of strict epimorphisms (also finite).
This also shows that the composition of two strict epimorphisms is not necessarily a strict epimorphism.
563fga3.i-a.2.cfga3.i-a.2.c.xmlA.2.cfga3.i-a.2
If is an exact diagram of finite morphisms, then, for every flat morphism T \to S of prescheme, the diagram induced from by a change of base T \to S is again exact.
It thus follows that, if X and Y are S-preschemes, with X flat over S, then the following diagram of maps (where X' and Y' are the inverse images of X and Y over S', and X'' and Y'' are their inverse images over S'') is exact:
\operatorname {Hom} _S(X,Y) \to \operatorname {Hom} _{S'}(X',Y') \rightrightarrows \operatorname {Hom} _{S''}(X'',Y'').
In particular, if \mathcal {F} denotes the fibred category (over the category \mathcal {C} of preschemes) such that, for X \in \mathcal {C}, \mathcal {F}_X is the category of flat X-preschemes, then the diagram is \mathcal {F}-exact.
(This result becomes false if we do not impose the flatness hypothesis; in particular, a finite strict epimorphism is not necessarily a descent morphism).
We similarly see that is \mathcal {F}-exact if \mathcal {F} denotes the fibred category for which \mathcal {F}_X is the category of flat quasi-coherent sheaves on the prescheme X (here, again, the flatness hypothesis is essential).
In either case, the question of effectiveness of a gluing data (and, more specifically, of a descent data, when S''=S' \times _S S') on a flat object over S' is delicate (and its answer in many particular cases in one of the principal objects of these current articles).
The speaker does not know if, for every finite strict epimorphism S' \to S, every descent data on a flat quasi-coherent sheaf on S' is effective (even if we suppose that S is the spectrum of a local Artinian ring, and we restrict to locally free sheaves of rank 1).
More generally, let A be a ring, and A' an A-algebra (where everything is commutative) such that the diagram
A \to A' \rightrightarrows A' \otimes _A A'
is exact, which is equivalent to saying that the corresponding morphism S' \to S between the spectra of A' and A is an \mathcal {F}-descent morphism, where \mathcal {F} is the fibred category of flat quasi-coherent sheaves.
Let M' be a flat A'-module endowed with a descent data to A, i.e. with an isomorphism
\varphi \colon M' \otimes _A A' \xrightarrow { \sim } A' \otimes _A M'
of (A' \otimes _A A')-modules that satisfies conditions (i) and (ii) of (which we leave to the reader to write out in terms of modules).
Is this data effective (relative to the fibred category of flat quasi-coherent sheaves)?
Let M be the subset of M' consisting of the x' \in M' such that
\varphi (x' \otimes _A 1_{A'}) = 1_{A'} \otimes _A x',
which is a sub-A-module of M'.
The canonical injection M \to M' defines a homomorphism of A'-modules M \otimes _A A' \to M'.
The effectiveness of \varphi then implies the following: M is a flat A-module, and the above homomorphism is an isomorphism.562Remarkfga3.i-a.2.c-remarkfga3.i-a.2.c-remark.xmlfga3.i-a.2.c
In the above, we have imposed no flatness hypotheses on the morphisms of the diagram , and this obliges us, in order to have a technique of descent, to impose flatness hypotheses on the objects over S and S' that we consider.
In , we will impose a flatness hypothesis on \alpha \colon S' \to S, which will allow us to have a technique of descent for objects over S and S' that are no longer under any flatness hypotheses.
In any case, there is a flatness hypothesis involved somewhere.
This is one of the main reasons for the importance of the notion of flatness in algebraic geometry (whose role could not be visible when we restricted to base fields, over which everything, in fact, is flat!).
568fga3.i-a.3fga3.i-a.3.xmlApplication to étalementsA.3fga3.i-a
Let A be a local ring, and B a local algebra over A whose maximal ideal induces that of A.
We say that B is étalé over A (instead of "unramified", as used elsewhere) if it satisfies the following conditions:
B is flat over A; and
B/ \mathfrak {m}B is a separable finite extension of A/ \mathfrak {m}=k (where \mathfrak {m} denotes the maximal ideal of A).
If A and B are Noetherian, and k is algebraically closed, then this implies that the homomorphism \overline {A} \to \overline {B} between the completions that extends A \to B is an isomorphism.
A morphism f \colon T \to S of finite type is said to be étale at x \in T, or T is said to be étalé over S at x, if \mathscr { O } _x is étalé over \mathscr { O } _{f(x)};
f is said to be étale, or an étalement, or T is said to be étalé over S, if f is étale at all x \in T.
Note that, if S is locally Noetherian, then the set of points of T where f is étale is open, and Zariski's "main theorem" allows us to precisely state the structure of T/S in a neighbourhood around such a point (by an equation of well-known type).
If S is a scheme of finite type over the field of complex numbers, then there exists a corresponding analytic space \overline {S} (in the sense of Serre [Ser1956]), except for the fact that \overline {S} can have nilpotent elements in its structure sheaf, which changes nothing essential in [Ser1956].
We then easily see that f is an étalement if and only if \overline {f} \colon \overline {T} \to \overline {S} is an étalement, i.e. if every point of \overline {T} admits a neighbourhood on which \overline {f} induces an isomorphism onto an open subset of \overline {S}.
In particular, to every étale covering T of S (i.e. every finite étale morphism f \colon T \to S), there is a corresponding étale covering \overline {T} of \overline {S}, which is connected if and only if T is connected [Ser1956].
We can also easily see that, if T and T' are étale schemes over S, then the natural map
\operatorname {Hom} _S(T,T') \to \operatorname {Hom} _{ \overline {S}}( \overline {T}, \overline {T}'')
is bijective, i.e. the functor T \mapsto \overline {T} from the category of étale schemes over S to the category of analytic spaces over S is "fully faithful", and thus defines an equivalence between the first category and a subcategory of the second.
A theorem of Grauert–Remmert [GR1958] implies that, if S is normal, then we thus obtain an equivalence between the category of étale coverings of S and the category of (finite) étale coverings of S, i.e. every étale covering \mathscr { C } of \overline {S} is \overline {S}-isomorphic to some \overline {T}, where T is an étale covering of S.
We will show that the Grauert–Remmert theorem remains true without any normality hypotheses on S.
First let S' \to S be a finite strict epimorphism, and suppose that the theorem has been proven for S'; we will show that it holds for S.
Let \mathscr { C } be an étale covering of \overline {S}, and consider its inverse image \mathscr { C } ' over S', which corresponds to a coherent analytic sheaf \mathfrak {A}' of algebras on S' that is the inverse image of a sheaf of algebras \mathfrak {A} on \overline {S} defining \mathscr { C }.
By hypothesis, on S', \mathscr { C } ' comes from an étale covering T' of S', i.e. \mathfrak {A}' comes from a coherent sheaf of algebras \mathscr { A } ' on S'.
Also, \mathfrak {A}' is endowed with a canonical descent data with respect to \overline {S}' \to \overline {S}, i.e. with an isomorphism between its two inverse images on \overline {S}' \times _{ \overline {S}} \overline {S}'= \overline {S' \times _SS'} (satisfying conditions (i) and (ii)), and this isomorphism comes from, by what has already been said, an isomorphism between the corresponding algebraic sheaves, i.e. from a descent data on \mathscr { A } ' with respect to S' \to S.
We can easily show that the latter is effective (since it is effective on \mathfrak {A}', and the effectiveness of a descent data, as described explicitly in the previous section, is something that can be checked locally on the completions of the modules that are involved).
From this, we obtain a coherent sheaf of algebras \mathscr { A } on S that defines a covering T of S, and this is the desired covering.
The above result then obviously holds true if S' \to S is just a composition of a finite number of finite strict epimorphisms, i.e. is just an arbitrary finite epimorphism (by the factorisation result stated in ).
It thus follows that the Grauert–Remmert theorem holds true if S is a reduced scheme, i.e. such that \mathscr { O } _S has no nilpotent elements, as we can see by introducing its normalisation S'.
We can easily pass to the general case.
A completely analogous argument, again using the factorisation result for finite strict epimorphisms, and the "formal" nature of the effectiveness of descent data, allows us to prove the following result:
let S be a locally Noetherian prescheme, and let S' \to S be a finite, surjective, and radicial morphism (or, equivalently, a morphism of finite type such that, for every T over S, the morphism T'=S' \times _S T \to T is a homeomorphism, which we can also express by saying that S' \to S is a universal homeomorphism).
For every T étalé over S, consider its inverse image T'=T \times _S S', which is étalé over S'.
Then the functor T \mapsto T' is an equivalence between the category of preschemes T that are étalé over S and the category of preschemes T' that are étalé over S'.
(We use the bijectivity of
\operatorname {Hom} _S(T_1,T_2) \to \operatorname {Hom} _{S'}(T'_1,T'_2)
for preschemes T_1 and T_2 that are étalé over S, which can be proven directly without difficulty. We also use the fact that the stated theorem is true if S'=(S, \mathscr { O } _S/ \mathscr { J } ), where \mathscr { J } is a nilpotent coherent sheaf of ideals of \mathscr { O } _S, cf. [Mur1958, Lemma 6]).
Note also that we do not suppose here that the T in question are finite over S.
This result implies, in particular, that the morphism S' \to S induces an isomorphism between the fundamental group of S' and the fundamental group of S ("topological invariance of the fundamental group of a prescheme").
586fga3.i-a.4fga3.i-a.4.xmlRelations to 1-cohomologyA.4fga3.i-a569fga3.i-a.4.afga3.i-a.4.a.xmlA.4.afga3.i-a.4
Let \mathcal {C} be a category such that the product of any two objects always exists, and let T \in \mathcal {C}.
For every finite set I \neq \varnothing, we can consider T^I, and so we obtain a covariant functor from the category of non-empty finite sets to the category \mathcal {C}, i.e. what we can call a simplicial object of \mathcal {C}, denoted by K_T.
This object depends covariantly on T;
also, if u and v are morphisms T \to T', then the corresponding morphisms K_T \to K_{T} are homotopic.
We say that T dominates T' if \operatorname {Hom} (T,T') \neq \varnothing, and this gives an (upward) directed preorder on \mathcal {C}.
It follows from the above that, if T dominates T', then there exists a canonical class (up to homotopy) of homomorphisms of simplicial objects K_T \to K_{T'};
in particular, if K_T and K_{T'} are such that T and T' dominate one another, then K_T and K_{T'} are homotopically equivalent.
Now let F be a (contravariant, to be clear) functor from \mathcal {C} to an abelian category \mathcal {C}'.
Then
C^ \bullet (T,F) = F(K_T)
is a cosimplicial object of \mathcal {C}', and thus defines, in a well-known way, a (cochain) complex in \mathcal {C}', of which we can take the cohomology:
\operatorname {H} ^ \bullet (T,F) = \operatorname {H} ^ \bullet (C^ \bullet (T,F)) = \operatorname {H} ^ \bullet (F(K_T))
(we may write a subscript "\mathcal {C}" on the \operatorname {H} ^ \bullet if there is any possibility for confusion).
This is a cohomological functor in F, of which the variance for T varying follows from what has already been said about the K_T;
more precisely, for fixed F and varying T in \mathcal {C} (preordered by the domination relation), the \operatorname {H} ^ \bullet (T,F) form an inductive system of graded objects of \mathcal {C}';
in particular, if T and T' are such that each one dominates the other, then \operatorname {H} ^ \bullet (T,F) and \operatorname {H} ^ \bullet (T',F) are canonically isomorphic.
Suppose that \mathcal {C} has all fibre products.
Then we can, for fixed S \in \mathcal {C}, apply the above to the category \mathcal {C}_S of objects of \mathcal {C} over S;
we then write C^ \bullet (T/S,F) and \operatorname {H} ^ \bullet (T/S,F) instead of C^ \bullet (T,F) and \operatorname {H} ^ \bullet (T,F) if we wish to make clear that we are working in the category \mathcal {C}_S;
then C^ \bullet (T/S,F) is a cochain complex in \mathcal {C}' that, in degree n, is equal to F(T \times _S T \times _S \ldots \times _S T) (where there are n+1 factors T).
Note that, as per usual, we can define \operatorname {H} ^0(T/S,F) without assuming the category \mathcal {C}' to be abelian:
it is the kernel (), if it exists, of the pair (F(p_1),F(p_2)) of morphisms
F(T) \to F(T \times _S T)
corresponding to the two projections p_1,p_2 \colon T \times _S T \to T.
In particular, we then have the natural morphism (called the augmentation)
F(S) \to \operatorname {H} ^0(T/S,F)
which is an isomorphism in nice cases (in particular, in the case where T \to S is a strict epimorphism and F is "left exact").
Similarly, if F takes values in the category of groups in a category \mathcal {C}'', then we can also define \operatorname {H} ^1(T/S,F);
in the case where \mathcal {C}'' is the category of sets (i.e. when F takes values in the category of non-necessarily-commutative groups), \operatorname {H} ^1(T,F) is the quotient of the subgroup Z^1(T/S,F) of C^1(T/S,F) = F(T \times _S T) consisting of the g such that
F(p_{31})(g) = F(p_{32})(g) F(p_{21})(g)
by the group with operators F(T) acting on C^1(T/S,F), and thus, in particular, on the subset Z^1(T/S,F), by
\rho (g') \cdot g = F(p_2)(g') g F(p_1)(g')^{-1}. 570fga3.i-a.4.bfga3.i-a.4.b.xmlA.4.bfga3.i-a.4
For example, let \mathcal {F} be a fibred category with base \mathcal {C}.
Let \xi , \eta \in \mathcal {F}_S, and, for all S' over S, let
F_{ \xi , \eta }(S') = \operatorname {Hom} ( \xi \times _S S', \eta \times _S S').
Then F_{ \xi , \eta } is a contravariant functor from \mathcal {C}_S to the category of sets.
With this setup, saying that the augmentation morphism
F_{ \xi , \eta }(S) \to \operatorname {H} ^0(S'/S,F_{ \xi , \eta })
is an isomorphism for every pair of elements \xi , \eta \in \mathcal {F}_S implies that \alpha \colon S' \to S is an \mathcal {F}-descent morphism ().
572fga3.i-a.4.cfga3.i-a.4.c.xmlA.4.cfga3.i-a.4
Similarly, for \xi \in \mathcal {F}_S and any object S' of \mathcal {C} over
G_ \xi (S') = \operatorname {Aut} ( \xi \times _S S'),
we thus define a contravariant functor G_ \xi from \mathcal {C}_S to the category of groups.
With this setup, we claim that Z^1(S'/S,G) is canonically identified with the set of descent data on \xi '= \xi \times _S S' with respect to S' \to S (), and that \operatorname {H} ^1(S'/S,G) can be identified with the set of isomorphism classes of objects of \mathcal {F}_{S'} endowed with a descent data relative to \alpha \colon S' \to S that are isomorphic, as objects of \mathcal {F}_{S'}, to \xi '= \xi \times _S S'.
Then, _if \alpha \colon S' \to S is an \mathcal {F}-descent morphism_ (cf. ), _then \operatorname {H} ^1(S'/S,G) contains as a subset the set of isomorphism classes of objects \eta of \mathcal {F}_S such that \eta \times _S S' is isomorphic (in \mathcal {F}_{S'}) to \xi \times _S S'_;
further, this inclusion is the identity if and only if every descent data on \xi '= \xi \times _S S' with respect to \alpha \colon S' \to S is effective.
(This will be the case, in particular, if \alpha \colon S' \to S is a strict S-descent morphism).
571Remarkfga3.i-a.4.c-remarkfga3.i-a.4.c-remark.xmlfga3.i-a.4.c
The cochain complexes of the form C^ \bullet (T/S,F) contain, as particular cases, the majority of standard known complexes (that of Čech cohomology, of group cohomology, etc.), and play an important role in algebraic geometry (notably in the "Weil cohomology" of preschemes).
574fga3.i-a.4.dfga3.i-a.4.d.xmlA.4.dfga3.i-a.4573Examplefga3.i-a.4-example-1fga3.i-a.4-example-1.xml1fga3.i-a.4.d
Let S' be an object over S \in \mathcal {C}, and let \Gamma be a group of automorphisms of S' such that S' is "formally \Gamma-principal over S", i.e. such that the natural morphism
\Gamma \times S' \to S' \times _S S'
(where \Gamma \times S' denotes the direct sum of \Gamma copies of S') is an isomorphism.
(We suppose that all the necessary direct sums exist in \mathcal {C}).
Let F be a contravariant functor from \mathcal {C} to the category of abelian groups.
Then C^ \bullet (S'/S,F) is canonically isomorphic to the simplicial group C^ \bullet ( \Gamma ,F(S')) of standard homogeneous cochains, and so \operatorname {H} ^ \bullet (S'/S,F) is canonically isomorphic to \operatorname {H} ^ \bullet ( \Gamma ,F(S')).
585fga3.i-a.4.efga3.i-a.4.e.xmlA.4.efga3.i-a.4583Examplefga3.i-a.4-example-2fga3.i-a.4-example-2.xml2fga3.i-a.4.e
Let \mathcal {C} be the category of preschemes.
We denote by \operatorname {G_a} (for "additive group") the contravariant functor from \mathcal {C} to the category of abelian groups, defined by
\operatorname {G_a} (X) = \operatorname {H} ^0(X, \mathscr { O } _X).
We similarly define the functor \operatorname {G_m} (for "multiplicative group") by
\operatorname {G_m} (X) = \operatorname {H} ^0(X, \mathscr { O } _X)^ \times
(i.e. the group of invertible elements of the ring \operatorname {H} ^0(X, \mathscr { O } _X)), and, more generally, the functor \operatorname {GL} (n) (for "linear group of order n") by
\operatorname {GL} (n)(X) = \operatorname {GL} (n, \operatorname {H} ^0(X, \mathscr { O } _X)),
which is a functor from \mathcal {C} to the category of (not-necessary-commutative, if n>1) groups;
for n=1 we recover \operatorname {G_m}.
We can also think of \operatorname {GL} (n) as an automorphism functor (cf. ) by considering the fibred category \mathcal {F} with base \mathcal {C} such that \mathcal {F}_X is the category of locally free sheaves on X, for X \in \mathcal {C}, since then \operatorname {GL} (n)(X)= \operatorname {Aut} _{ \mathcal {F}_X}( \mathscr { O } _X^n).
By , it follows that, if \alpha \colon S' \to S is an \mathcal {F}-descent morphism (cf. ), then \operatorname {H} ^1(S'/S, \operatorname {GL} (n)) contains the set of isomorphism classes of locally free sheaves on S whose inverse image on S' is isomorphic to \mathscr { O } _{S'}^n, and this inclusion is an equality if and only if every descent data on \mathscr { O } _{S'}^n (with respect to \alpha \colon S' \to S) is effective.
If S is the spectrum of a local ring, then this implies that \operatorname {H} ^1(S'/S, \operatorname {GL} (n))=(e), since every locally free sheaf on S is then trivial.
We note that the following conditions concerning a morphism \alpha \colon S' \to S are equivalent:
The augmentation homomorphism \operatorname {H} ^0(S, \mathscr { O } _S) = \operatorname {G_a} (S) \to \operatorname {H} ^0(S'/S, \operatorname {G_a} ) is an isomorphism.
\alpha \colon S' \to S is an \mathcal {F}-descent morphism (where \mathcal {F} is the fibred category over \mathcal {C} described above).
\alpha \colon S' \to S is a strict epimorphism (cf. ).
Now suppose that S= \operatorname {Spec} (A) and S'= \operatorname {Spec} (A');
then
C^n(S'/S, \operatorname {G_a} ) = C^n(A'/A, \operatorname {G_a} ) = \underbrace {A' \otimes _A A' \otimes _A \ldots \otimes _A A'}_{n+1 \text { copies of }A'}
with the coboundary operator C^n(A'/A, \operatorname {G_a} ) \to C^{n+1}(A'/A, \operatorname {G_a} ) being the alternating sum of the face operators
\partial _i(x_0 \otimes x_1 \otimes \ldots \otimes x_n) = x_0 \otimes \ldots \otimes x_{i-1} \otimes1 _{A'} \otimes x_i \otimes \ldots \otimes x_n.
Similarly, C^n(S'/S, \operatorname {G_m} )=C^n(A'/A, \operatorname {G_m} ) can be identified with ( \bigotimes _A^{n+1}A')^ \times, with the simplicial operations for C^ \bullet (A'/A, \operatorname {G_m} ) being induced by those in C^ \bullet (S'/S, \operatorname {G_a} ).
We can write down the simplicial operations for C^ \bullet (A'/A, \operatorname {GL} (n)) in the same explicit manner.
In all the cases known to the speaker, \operatorname {H} ^i(A'/A, \operatorname {G_a} )=0 for i>0, and, if A is local, then \operatorname {H} ^1(A'/A, \operatorname {G_m} )=0, and, more generally, \operatorname {H} ^1(A'/A, \operatorname {GL} (n))=(e) (if S' \to S is an \mathcal {F}-descent morphisms, i.e. if the diagram A \to A' \rightrightarrows A' \otimes _A A' is exact, then compare this with ).
We note that Hilbert's "Theorem 90" is exactly the fact that \operatorname {H} ^1(S'/S, \operatorname {G_m} )=0 if A is a field and A' is a finite Galois extension of A (cf. ), and we can also express it by saying that, in the case in question, S' \to S is a strict descent morphisms for the fibred category of locally free sheaves of rank 1.
This latter statement is the one that is most suitable to generalise Hilbert's theorem, by varying the hypotheses both on the morphism S' \to S and on the quasi-coherent sheaves in question.
Finally, we note that, for a local Artinian A with maximal ideal \mathfrak {m}, and an A-algebra A' (where we denote, for any integer k>0, the ring A/ \mathfrak {m}^{k+1} (resp. A'/ \mathfrak {m}^{k+1}A') by A_k (resp. A'_k)), the following properties are equivalent:
\operatorname {H} ^1(A'_k/A_k, \operatorname {G_a} )=0 for all k.
\operatorname {H} ^1(A'_k/A_k, \operatorname {G_m} )=0 for all k.
\operatorname {H} ^1(A'_k/A_k, \operatorname {GL} (n))=(e) for all k and all n.
If S' \to S is a strict epimorphism, then the above conditions imply that it is a strict descent morphism for free modules (not necessarily of finite type) over A'.
584Remarkfga3.i-a.4.e-remarkfga3.i-a.4.e-remark.xmlfga3.i-a.4.e
The definition of the groups \operatorname {H} ^i(S'/S, \operatorname {G_m} ) in the case where S (resp. S') is a scheme over the field A (resp. A') is due to Amitsur.
The group \operatorname {H} ^2(S'/S, \operatorname {G_m} ) is particularly interesting as a "global" variant of the Brauer group, for which we can refer to [GD1960, VII].
624fga3.i-bfga3.i-b.xmlDescent by faithfully flat morphismsBfga3.i603fga3.i-b.1fga3.i-b.1.xmlStatement of the descent theoremsB.1fga3.i-b588Definitionfga3.i-b.1-definition-1.1fga3.i-b.1-definition-1.1.xml1.1fga3.i-b.1
A morphism \alpha \colon S' \to S of prescheme is said to be flat if \mathscr { O } _{x'} is a flat module over the ring \mathscr { O } _{ \alpha (x')} for all x' \in S' (i.e. if \mathscr { O } _{x'} \otimes _{ \mathscr { O } _{ \alpha (x')}}M is an exact functor in the \mathscr { O } _{ \alpha (x')}-module M).
A morphism is said to be faithfully flat if it is flat and surjective.
For example, if S= \operatorname {Spec} (A) and S'= \operatorname {Spec} (A'), then S' is flat over S if and only if A' is a flat A-module, and S' is faithfully flat over S if and only if A' is a faithfully flat A-module (i.e. if and only if A' \otimes _A M is an exact and faithful functor in the A-module M);
this also implies, in the terminology of Serre [Ser1956], that the pair (A,A') is flat.
If S' is faithfully flat over S, then the inverse image functor of quasi-coherent sheaves on S is exact and faithful;
in other words, for a sequence of homomorphisms of quasi-coherent sheaves on S to be exact, it is necessary and sufficient that its inverse image on S' be exact (in particular, for a homomorphism of quasi-coherent sheaves on S to be a monomorphism (resp. an epimorphism, resp. an isomorphism), it is necessary and sufficient that its inverse image on S' be a monomorphism (resp. an epimorphism, resp. an isomorphism)).
This property holds true if we restrict to an arbitrary open subset of S', and then characterise faithfully flat morphisms in this form.
589Definitionfga3.i-b.1-definition-1.2fga3.i-b.1-definition-1.2.xml1.2fga3.i-b.1
A morphism \alpha \colon S' \to S is said to be quasi-compact if the inverse image of every quasi-compact open subset U of S is quasi-compact (i.e. a finite union of affine open subsets).
It evidently suffices to verify this property for the affine open subsets of S.
For example, an affine morphism (i.e. a morphism such that the inverse image of an affine open subset is affine) is quasi-compact.
The class of flat (resp. faithfully flat, resp. quasi-compact) morphisms is stable under composition and by "base extension", and of course contains all isomorphisms.
590Theoremfga3.i-b.1-theorem-1fga3.i-b.1-theorem-1.xml1fga3.i-b.1
Let \alpha \colon S' \to S be a morphism of preschemes that is faithfully flat and quasi-compact.
Then \alpha is a strict descent morphism (cf. ) for the fibred category \mathcal {F} of quasi-coherent sheaves (cf. §A, Example 2).
This statement implies two things:
If \mathcal {F} and \mathscr { G } are quasi-coherent sheaves on S, and \mathcal {F}' and \mathscr { G } ' their inverse images on S', then the natural homomorphism
\operatorname {Hom} ( \mathcal {F}, \mathscr { G } ) \to \operatorname {Hom} ( \mathcal {F}', \mathscr { G } ')
is a bijection from the left-hand side to the subgroup of the right-hand side consisting of homomorphisms \mathcal {F}' \to \mathscr { G } ' that are compatible with the canonical descent data on these sheaves, i.e. whose inverse images under the two projections of S''=S' \times _S S' to S' give the same homomorphism \mathcal {F}'' \to \mathscr { G } ''.
Every quasi-coherent sheaf \mathcal {F}' on S' endowed with a descent data with respect to the morphism \alpha \colon S' \to S (cf. ) is isomorphic (endowed with this data) to the inverse image of a quasi-coherent sheaf \mathcal {F} on S.
Setting \mathcal {F}= \mathscr { O } _S in (i), we obtain:
594Corollaryfga3.i-b.1-corollary-1fga3.i-b.1-corollary-1.xml1fga3.i-b.1
Let \mathscr { G } be a quasi-coherent sheaf on S, with \mathscr { G } ' and \mathscr { G } '' denoting its inverse images on S' and S''=S' \times _S S' (respectively), and let p_1 and p_2 be the two projections from S'' to S.
Then the diagram of maps of sets
\Gamma ( \mathscr { G } ) \xrightarrow { \alpha ^*} \Gamma ( \mathscr { G } ') \overset { p_1^* }{ \underset { p_2^* }{ \rightrightarrows }} \Gamma ( \mathscr { G } '')
is exact (cf. ).
Also, the combination of (i) and (ii) with gives:
595Theoremfga3.i-b.1-corollary-2fga3.i-b.1-corollary-2.xml2fga3.i-b.1
Let \mathscr { G } be as in .
Then there is a bijective correspondence between quasi-coherent subsheaves of \mathscr { G } and quasi-coherent subsheaves of \mathscr { G } ' whose inverse images on S'' under the two projections p_1 and p_2 give the same subsheaf of \mathscr { G }.
Of course, we have an equivalent statement in terms of quotient sheaves.
As we have already seen in , should be thought of as a generalisation of Hilbert's "Theorem 90", and implies, as particular cases, various formulations in terms of 1-cohomology.
For the proof, we can easily reduce to the case where S= \operatorname {Spec} (A) and S'= \operatorname {Spec} (A'), and, for (i), we can easily restrict to proving , i.e. the exactness of the diagram
M = A \otimes _A M \to A' \otimes _A M \rightrightarrows A' \otimes _A A' \otimes _A M
for every A-module M, which follows from the more general lemma:
597Lemmafga3.i-b.1-lemma-1.1fga3.i-b.1-lemma-1.1.xml1.1fga3.i-b.1
Let A' be a faithfully flat A-algebra.
Then, for every A-module M, the M-augmented complex C^ \bullet (A'/A, \operatorname {G_a} ) \otimes _A M (cf. ) is a resolution of M.
596Prooffga3.i-b.1-lemma-1.1
It suffices to prove that the augmented complex induced from the above by extension of the base A to A' satisfies the same conclusions.
This leads to proving the statement when we replace A by A', and A' by A' \otimes _A A', and so we can restrict to the case where there exists an A-algebra homomorphism A' \to A (or, in geometric terms, the case where S' over S admits a section).
In this case, the claim follows from the generalities of .
We note, in passing, the following corollary, which generalises a well-known statement in Galois cohomology (compare with ):
598Corollaryfga3.i-b.1-lemma-1.1-corollaryfga3.i-b.1-lemma-1.1-corollary.xmlfga3.i-b.1
If A' is faithfully flat over A, then \operatorname {H} ^0(A'/A, \operatorname {G_a} )=A, and \operatorname {H} ^i(A'/A, \operatorname {G_a} )=0 for i \geqslant1.
To prove part (ii) of , we proceed, as for (i), by restricting to the case where S' over S admits a section, where the result then follows from (i) (cf. ).
We can evidently vary and its corollaries ad libitum by introducing various additional structures on the quasi-coherent sheaves (or systems of sheaves) in question.
For example, the data on S of a quasi-coherent sheaf of commutative algebras "is equivalent to" the data on S' of such a sheaf endowed with a descent data relative to \alpha \colon S' \to S.
Taking into account the functorial correspondence between such quasi-coherent sheaves on S and affine preschemes over S, we obtain the second claim of the following theorem:
599Theoremfga3.i-b.1-theorem-2fga3.i-b.1-theorem-2.xml2fga3.i-b.1
Let \alpha \colon S' \to S be as in .
Then \alpha is a (non-strict, in general) descent morphism (cf. §A, Definition 2.4), and it is a strict descent morphism for the fibred category of affine schemes over preschemes (cf. §A, Definition 1.7).
The first claim of the theorem implies this:
let X and Y be preschemes over S, with X' and Y' their inverse images over S, and X'' and Y'' their inverse images over S''=S' \times _S S';
then the diagram of natural maps
\operatorname {Hom} _S(X,Y) \xrightarrow { \alpha ^*} \operatorname {Hom} _{S'}(X',Y') \overset { p_1^* }{ \underset { p_2^* }{ \rightrightarrows }} \operatorname {Hom} _{S''}(X'',Y'')
is exact, i.e. \alpha ^* is a bijection from \operatorname {Hom} _S(X,Y) to the subset of \operatorname {Hom} _{S'}(X',Y') consisting of homomorphisms that are compatible with the canonical descent data on X' and Y' (i.e. whose inverse images under the two projections from S'' to S' are equal).
This follows easily from and , if we restrict to Y being affine over S;
in the general case, we need to combine with the following result:
600Lemmafga3.i-b.1-lemma-1.2fga3.i-b.1-lemma-1.2.xml1.2fga3.i-b.1
Let \alpha \colon S' \to S be a faithfully flat and quasi-compact morphism.
Then S can be identified with a topological quotient space of S', i.e. every subset U of S such that \alpha ^{-1}(U) is open, is open.
To complete , we must give effectiveness criteria for a descent data on an S'-prescheme X' (in the case where X' is not assumed to be affine over S').
Note first of all that such a descent data is not necessarily effective, even if S is the spectrum of a field k, S' the spectrum of a quadratic extension k' of k, and S'' a proper algebraic scheme of dimension 2 over S' (as we can see, due to Serre, by using the non-projective surface of Nagata).
For a descent data on X'/S' with respect to \alpha \colon S' \to S (assumed to be faithfully flat and quasi-compact) to be effective, it is necessary and sufficient that X' be a union of open subsets X'_i that are affine over S' and "stable" under the descent data on X'.
This is certainly the case (for any X'/S' and any descent data on X') if the morphism \alpha \colon S' \to S is radicial (i.e. injective, and with radicial residual extensions).
We can also show that this is the case if \alpha \colon S' \to S is finite, and every finite subset of X' that is contained in a fibre of X' over S is also contained in an open subset of X' that is affine over S (this is the Weil criterion).
It is, in particular, the case if X'/S' is quasi-projective, and, in this case, we can show that the "descended" prescheme X/S is also quasi-projective (and projective if X'/S' is projective).
In summary:
601Theoremfga3.i-b.1-theorem-3fga3.i-b.1-theorem-3.xml3fga3.i-b.1
Let \alpha \colon S' \to S be faithfully flat and quasi-compact morphism of preschemes.
If \alpha is radicial, then it is a strict descent morphism.
If \alpha is finite, then it is a strict descent morphism with respect to the fibred category of quasi-projective (or projective) preschemes over preschemes.
602Remarkfga3.i-b.1-remarkfga3.i-b.1-remark.xmlfga3.i-b.1
I do not know if, in the second claim above, the hypothesis that \alpha be finite is indeed necessary;
we can prove that, in any case, we can "formally" replace it by the following, seemingly weaker, hypothesis:
for every point of S there exists an open neighbourhood U, a finite and faithfully flat U' over U, and an S-morphism from U' to S'.
A type of case that is not covered by the above is that where S= \operatorname {Spec} (A) and S'= \operatorname {Spec} ( \overline {A}), with A a local Noetherian ring and \overline {A} its completion;
or even that where S' is quasi-finite over S (i.e. locally isomorphic to an open subset of a finite S-scheme) but not finite.
In these two cases, the speaker also does not know the answer to the following question:
let X be an S-scheme such that X'=X \times _S S' is projective over S';
is it then true that X is projective over S?
[Comp.]
A morphism S' \to S that is quasi-finite, étale, surjective, or a morphism of the form \operatorname {Spec} ( \overline {A}) \to \operatorname {Spec} (A), is not always a strict descent morphism, even if A is the local ring of an algebraic curve over an algebraically closed field k and S= \operatorname {Spec} (A).
We can thus find a proper simple morphism f \colon X \to S that makes X into a principal E-bundle over S, with E an elliptic curve, such that f' \colon X' \to S' is projective, but f is not projective.
So this is also an example of a homogeneous principal bundle that is non-isotrivial under an abelian scheme.
604fga3.i-b.2fga3.i-b.2.xmlApplication to the descent of certain properties of morphismsB.2fga3.i-b
Let P be a class of morphisms of preschemes.
Let \alpha \colon S \to S' be a morphism of preschemes, and let f \colon X \to Y be a morphism of S-preschemes, with f' \colon X' \to Y' the inverse image of f under \alpha.
We can then ask if the relation "f' \in P" implies that "f \in P".
It appears that the answer is affirmative in many important cases, if we suppose that \alpha is faithfully flat and quasi-compact (this latter hypothesis sometimes being overly strong).
We can see this directly without difficulty if P is the class of surjective (resp. radicial) morphisms (with these two cases following from the surjectivity of \alpha), or flat (resp. faithfully flat, resp. simple) morphisms (with these three cases following from the faithful flatness of \alpha), or morphisms of finite type.
Using , , and , we see that it is also true if P is one of the following classes:
isomorphisms, open immersions, closed immersions, immersions (if f is of finite type, and Y is locally Noetherian), affine morphisms, finite morphisms, quasi-finite morphisms, open morphisms, closed morphisms, homeomorphisms, separated morphisms, or proper morphisms.
The only important case not covered here is that of projective or quasi-projective morphisms, which has already been discussed in the remark in .
618fga3.i-b.3fga3.i-b.3.xmlDecent by finite faithfully flat morphismsB.3fga3.i-b
Let \alpha \colon S' \to S be a finite morphism, corresponding to a sheaf of algebras \mathscr { A } ' on S that is locally free and of finite type as a sheaf of modules, and everywhere non-zero.
Then \alpha is a faithfully flat and quasi-compact morphism, to which we can thus apply the above results.
The data of a quasi-coherent sheaf \mathcal {F}' on S' is equivalent to the data of the quasi-coherent sheaf \alpha _*( \mathcal {F}') on S endowed with its \mathscr { A } '-modules structure (noting that \mathscr { A } '= \alpha _*( \mathscr { O } _{S'})).
For simplicity, we also denote this sheaf on S by \mathcal {F}'.
The two inverse images p_i^*( \mathcal {F}') of \mathcal {F}' on S' \times _S S' similarly correspond to the quasi-coherent sheaves of ( \mathscr { A } ' \otimes _{ \mathscr { O } _S} \mathscr { A } ')-modules \mathcal {F}' \otimes _{ \mathscr { O } _S} \mathscr { A } ' and \mathscr { A } ' \otimes _{ \mathscr { O } _S} \mathcal {F}'.
The data of an (S' \times _S S')-homomorphism from the former to the latter is equivalent to the data of a homomorphism of ( \mathscr { A } ' \otimes \mathscr { A } ')-modules, and, taking into account the fact that \mathscr { A } ' is locally free, this is equivalent to the data of a homomorphism of ( \mathscr { A } ' \otimes \mathscr { A } ')-modules:
\mathscr { U } = \mathscr {H} \kern -2.5pt \mathit {om} _{ \mathscr { O } _S}( \mathscr { A } ', \mathscr { A } ') = \mathscr { A } ' \otimes \check { \mathscr { A } }' \to \mathscr {H} \kern -2.5pt \mathit {om} _{ \mathscr { O } _S}( \mathcal {F}', \mathcal {F}')
i.e. to the data, for every section \xi of \mathscr { U } over an open subset V, of a homomorphism of \mathscr { O } _S-modules T_ \xi \colon \mathcal {F}'|V \to \mathcal {F}'|V that satisfies the conditions
605Equationfga3.i-b.3-equation-3.1fga3.i-b.3-equation-3.1.xml3.1fga3.i-b.3 \begin {aligned} T_{f \xi }(x) &= fT_ \xi (x), \\ T_{ \xi f}(x) &= T_ \xi (fx), \end {aligned} \tag{3.1}
where f and x are (respectively) sections of \mathscr { A } ' and \mathcal {F}' over an open subset of S that is contained inside V.
Conditions (i) and (ii) of a descent data (cf. ) can then be written (respectively) as
606Equationfga3.i-b.3-equation-3.2fga3.i-b.3-equation-3.2.xml3.2fga3.i-b.3 T_{1_U}(x) = x \qquad \text {i.e. }T_{1_U}= \operatorname {id} _{ \mathcal {F}'} \tag{3.2}
607Equationfga3.i-b.3-equation-3.3fga3.i-b.3-equation-3.3.xml3.3fga3.i-b.3 T_{ \xi \eta } = T_ \xi T_ \eta . \tag{3.3}
In other words, a descent data on \mathcal {F}' is equivalent to a representation of the sheaf \mathscr { U } = \mathscr {H} \kern -2.5pt \mathit {om} _{ \mathscr { O } _S}( \mathscr { A } ', \mathscr { A } ') of \mathscr { O } _S-algebras in the sheaf \mathscr {H} \kern -2.5pt \mathit {om} _{ \mathscr { O } _S}( \mathcal {F}', \mathcal {F}') of \mathscr { O } _S-algebras that satisfies the two linearity conditions ().
If we have a pairing of quasi-coherent sheaves on S':
\mathcal {F}'_1 \times \mathcal {F}'_2 \to \mathcal {F}'_3
(that we can think of as a pairing of sheaves of \mathscr { A } '-modules on S), and gluing data on the \mathcal {F}'_i defined by homomorphisms T_i \colon \mathscr { U } \to \mathscr {H} \kern -2.5pt \mathit {om} _{ \mathscr { O } _S}( \mathcal {F}'_i, \mathcal {F}'_i) (for i=1,2,3), then these data are equivalent to the given pairing, in the evident meaning of the phrase, if and only if the following condition is satisfied:
For every section \xi of \mathscr { U } over an open subset, and denoting by \Delta \xi = \sum \xi '_i \otimes _{ \mathscr { A } '} \xi ''_i the section of \mathscr { U } \otimes _{ \mathscr { A } '} \mathscr { U } (where \mathscr { U } is considered as an \mathscr { A } '-module with its left structure) defined by the formula
\xi \cdot (fg) = \sum _i \xi '_i(f) \xi ''_i(g)
(where f and g are sections of \mathscr { A } ' over a smaller open subset), we have the formula
608Equationfga3.i-b.3-equation-3.4fga3.i-b.3-equation-3.4.xml3.4fga3.i-b.3 T_ \xi ^{(3)}(x \cdot y) = \sum _i T_{ \xi '_i}^{(1)}x \cdot T_{ \xi ''_i}^{(2)}y \tag{3.4}
for every pair (x,y) of sections of \mathscr { A } ' over a smaller subset.
(We can express this property by saying that the homomorphisms T^{(i)} are compatible with the diagonal map of \mathscr { U }, with respect to the given pair).
In particular, , , , and allow us to understand, in terms of representations of algebras via diagonal maps, the descent data on a quasi-coherent sheaf of algebras on S', and thus also (by restricting to commutative algebras) the descent data on an affine S'-scheme.
From here, we obtain an analogous interpretation of descent data on an arbitrary S'-prescheme X':
the data of such an X' is equivalent to the data of a prescheme X' over S endowed with a homomorphism of \mathscr { O } _S-algebras
\mathscr { A } ' \to \mathscr { O } _{X'},
and a descent data on X' is equivalent to the data of a sheaf homomorphism
\mathscr { U } \to \mathscr {H} \kern -2.5pt \mathit {om} _{h^{-1}( \mathscr { O } _S)}( \mathscr { O } _{X'}, \mathscr { O } _{X'})
that is compatible with the morphism h \colon X' \to S' and that satisfies the conditions analogous to , , , and above.
609Examplefga3.i-b.3-example-1fga3.i-b.3-example-1.xmlWeil's example1fga3.i-b.3
Suppose that S'/S is a Galois étale covering with Galois group \Gamma (cf. and ).
Then a descent data on a quasi-coherent sheaf \mathcal {F}' on S' (resp. on an S'-prescheme X') is equivalent to the data of a representation of \Gamma by automorphisms of (S', \mathcal {F}') (resp. of (S',X')) that is compatible with the action of \Gamma on S'.
This result is "formal", i.e. it can be proven in terms of categories, but, from the point of view of this section, we also obtain the explicit structure of \mathscr { U } (endowed with its ring structure, the ring homomorphism \mathscr { A } ' \to \mathscr { U }, and the diagonal map), which is completely known thanks to the following result:
\mathscr { U } admits, as a left A'-module, a basis given by the sections of \mathscr { U } that correspond to elements of \Gamma.
611Examplefga3.i-b.3-example-2fga3.i-b.3-example-2.xmlCartier's example2fga3.i-b.3
Let p be a prime number, and suppose that p \mathscr { O } _S=0 (i.e. that \mathscr { O } _S is of characteristic p), that ( \mathscr { A } ')^p \subset \mathscr { O } _S= \mathscr { A } (i.e. that S'/S is radicial of height 1), and that the sheaf of algebras \mathscr { A } ' over \mathscr { A } locally admits a p-basis (i.e. a family (x_i) of sections such that \mathscr { A } ' is generated as an algebra by the x_i under the sole condition that x_i^p=0).
We suppose that the set of the i is finite, of cardinality n.
Let \mathfrak {C} be the sheaf of A-derivations of A', which is a locally free sheaf of rank n of A'-modules, and, furthermore, a sheaf of Lie p-algebras over \mathscr { A } (but not over \mathscr { A } ') that satisfies the following condition:
610Equationfga3.i-b.3-equation-3.5fga3.i-b.3-equation-3.5.xml3.5fga3.i-b.3-example-2 [X,fY] = X(f)Y + f[X,Y]. \tag{3.5} 613Lemmafga3.i-b.3-lemmafga3.i-b.3-lemma.xmlfga3.i-b.3\mathscr { U } = \mathscr {H} \kern -2.5pt \mathit {om} _{ \mathscr { O } _S}( \mathscr { A } ', \mathscr { A } ') is generated, as an \mathscr { O } _S-algebra endowed with an algebra homomorphism \mathscr { A } ' \to \mathscr { U }, by the sub-left-A'-module \mathfrak {C}, with the following additional relations:
612Equationfga3.i-b.3-equation-3.6fga3.i-b.3-equation-3.6.xml3.6fga3.i-b.3-lemma \begin {cases} Xf-fX &= X(f) \\ XY-YX &= [X,Y] \\ X^p &= X^{(p)}. \end {cases} \tag{3.6}
It follows from the above lemma that a descent data on the quasi-coherent sheaf \mathcal {F}' on S' is equivalent to the data, for all X \in \mathfrak {C}, of an \mathscr { O } _S-endomorphism \overline {X} of \mathcal {F}' that satisfies the following conditions:
614Equationfga3.i-b.3-equation-3.7fga3.i-b.3-equation-3.7.xml3.7fga3.i-b.3 \overline {fX} = f \overline {X} \tag{3.7}
615Equationfga3.i-b.3-equation-3.8fga3.i-b.3-equation-3.8.xml3.8fga3.i-b.3 \overline {X}(fx) = X(f)x + f \overline {X}(x) \tag{3.8}
616Equationfga3.i-b.3-equation-3.9fga3.i-b.3-equation-3.9.xml3.9fga3.i-b.3 \overline {[X,Y]} = [ \overline {X}, \overline {Y}] \tag{3.9}
617Equationfga3.i-b.3-equation-3.10fga3.i-b.3-equation-3.10.xml3.10fga3.i-b.3 \overline {X^{(p)}} = \overline {X}^p. \tag{3.10}
(This is what we may call a linear connection on \mathcal {F}, which is further flat and compatible with the p-th powers_).
We can similarly write down the notion of a descent data on an S'-prescheme X', with being replaced by the condition that the \overline {X} are derivations of \mathscr { O } _{X'}.
Since the morphism S' \to S is radicial, ensures that every such descent data is effective, and thus defines an S-prescheme X.
Note that we have not needed to impose any flatness, non-singular, or finiteness hypotheses on \mathcal {F}' or X'.
619fga3.i-b.4fga3.i-b.4.xmlApplication to rationality criteriaB.4fga3.i-b
Let X be an S-prescheme such that the direct image of \mathscr { O } _X on S is \mathscr { O } _S;
this property remains true for any flat base extension S' \to S.
If \mathcal {F} is an invertible sheaf (i.e. locally free of rank 1) on X, then there is a bijective correspondence between automorphisms of \mathcal {F} (identified with the invertible sections of \mathscr { O } _X) and invertible sections of \mathscr { O } _S.
So let s be a section of X over S;
we define a section of \mathcal {F} over s to be a section of the invertible sheaf s^*( \mathcal {F}) on S.
It follows from the above that, if \mathcal {F}_i (for i=1,2) are invertible sheaves on X, each endowed with a section over s, and if \mathcal {F}_1 and \mathcal {F}_2 are isomorphic, then there exists exactly one isomorphism from \mathcal {F}_1 to \mathcal {F}_2 that is compatible with the sections in question (i.e. sending the first to the second).
We also, independently of the section s, regard two invertible sheaves \mathcal {F}_1 and \mathcal {F}_2 on X as equivalent if every point of S has an open neighbourhood U such that the restrictions of \mathcal {F}_1 and \mathcal {F}_2 to X|U are isomorphic.
Then every invertible sheaf \mathcal {F} on X is equivalent to an invertible sheaf \mathcal {F}_1 endowed with a marked section over s (we take \mathcal {F}_1=Fs^*( \mathcal {F})^{-1}), and \mathcal {F}_1 is determined up to isomorphism.
In other words, the classification up to equivalence of invertible sheaves on X is the same as the classification up to isomorphism of invertible sheaves endowed with a marked section.
Since these properties remain true under flat extensions \alpha \colon S' \to S of the base (by replacing the section s with its inverse image s' under \alpha), we thus conclude, taking into account:
With the prescheme X/S being as above, and admitting a section s, let \alpha \colon S' \to S be a faithfully flat and quasi-compact morphism; let \mathcal {F}' be an invertible sheaf on X'=X \times _S S'.
For \mathcal {F}' to be equivalent to the inverse image on X' of an invertible sheaf \mathcal {F}' on X, it is necessary and sufficient that its inverse images p_1^*( \mathcal {F}') and p_2^*( \mathcal {F}') on X' \times _X X'=X \times _S(S' \times _S S') be equivalent.
If this is the case, then \mathcal {F} is determined up to equivalence.
(We then say that \mathcal {F}' is rational on S).
Considering this principle in the case where \alpha \colon S' \to S is as in and in the previous section, we recover the rationality criteria of Weil and of Cartier.
(We note that the authors restrict to the case where S and S' are spectra of fields;
a fortiori, S is then the spectrum of a local ring, and the equivalence relation introduced above is exactly the relation of being isomorphic).
The the first case, \mathcal {F}' is rational on S if and only if its images under \Gamma are all equivalent to \mathcal {F}'.
To express the rationality criterion in the second case, we consider, more generally, the diagonal morphism X' \to X''=X' \times _X X' of X'/X, with the corresponding sheaf of ideals \mathscr { I } on X' \times _X X', and the sheaf \mathscr { I } / \mathscr { I } ^2, which can be identified with its inverse image \Omega _{X'/X}^1 on X (the sheaf of 1-differentials of X' with respect to X).
Since the restrictions of the \mathcal {F}''_i=p_i( \mathcal {F}') (for i=1,2) to the diagonal are isomorphic (since they are both isomorphic to \mathcal {F}'), i.e. \mathcal {F}''_1( \mathcal {F}''_2)^{-1}= \mathcal {F}'' has a restriction to the diagonal which is trivial, it follows that the restriction of \mathcal {F}'' to (X'', \mathscr { O } _{X''}/ \mathscr { I } ^2) is given, up to isomorphism, by a well-defined element \xi of
\operatorname {H} ^1(X'', \mathscr { I } / \mathscr { I } ^2) = \operatorname {H} ^1(X', \Omega _{X'/X}^1).
Also, being precise, we have \Omega _{X'/X}^1= \Omega _{S'/S}^1 \otimes _{ \mathscr { O } _S} \mathscr { O } _X, and thus, if \Omega _{S'/S}^1 is locally free on S (as in the Cartier case), then \xi defines a section of \operatorname {R} ^1f'( \mathscr { O } _{X'}) \otimes \Omega _{S'/S}^1 on S' (called the Atiyah–Cartier class of the invertible sheaf \mathcal {F} on X'/S) whose vanishing is necessary and sufficient for the inverse images of \mathcal {F}' under the two projections of
(X'', \mathscr { O } _{X''}/ \mathscr { I } ^2) = X \times _S(S'', \mathscr { O } _{S''}/ \mathscr { J } ^2)
to X' to be equivalent (where \mathscr { J } is the sheaf of ideals on S''=S' \times _S S' defined by the diagonal morphism S' \to S' \times _S S').
This vanishing is thus trivially necessary for the inverse images of \mathcal {F}' on X''=X \times _S S'' itself to be equivalent, and thus also for \mathcal {F} to be equivalent to the inverse image of an invertible sheaf \mathcal {F} on X.
The Atiyah–Cartier class can also be understood as the obstruction to the existence, locally over S', of a connection of \mathcal {F}' relative to the derivations of X'/X, with such a connection further being determined when we know the derivations of \mathcal {F}' corresponding to the natural extensions of derivations of S'/S to X'.
From this, and the results of the previous section, we easily conclude that, in the case of the aforementioned , and when X/S admits a section, the vanishing of the Atiyah–Cartier class is also sufficient for \mathcal {F}' to be rational on S.
622fga3.i-b.5fga3.i-b.5.xmlApplication to the restriction of the base scheme to an abelian schemeB.5fga3.i-b
Let S be a prescheme.
We define an abelian scheme over S to be a simple proper scheme X over S whose fibres at the points x \in S are schemes of abelian varieties over the k(x).
Suppose that S is Noetherian and regular (i.e. that its local rings are regular), then we can show, using the connection theorem of Murre [Mur1958] (at least in the case "of equal characteristics", where the cited theorem is currently proven) that every rational section of X over S is everywhere defined (i.e. is a section) (which generalises a classical theorem of Weil).
It then follows, more generally, that, if X' is a simple scheme over S, then every rational S-map from X' to X is everywhere defined.
From this, we obtain the following, which generalises a result of Chow–Lang:
with S Noetherian and regular, and K denoting its ring of rational functions (a direct sum of fields), let X be an abelian scheme over K; if X is isomorphic to a K-scheme of the form X_0 \times _S \operatorname {Spec} (K), where X_0 is an abelian scheme over S, then X_0 is determined up to unique isomorphism.
Using the above uniqueness result, we see that the question of restriction of the base to X is local on S (and thus that it suffices to know how to do the restriction to \operatorname {Spec} ( \mathscr { O } _x), where x \in S).
In the same way, we see that, if S' \to S is a simple morphism of finite type, if Y' is the ring of rational functions of S', and if X \otimes _K K' is of the form X'_0 \times _{S'} \operatorname {Spec} (K'), then X'_0 is endowed with a canonical descent data with respect to \alpha.
Taking into account, we thus conclude:
620Propositionfga3.i-b.5-proposition-5.1fga3.i-b.5-proposition-5.1.xml5.1fga3.i-b.5
Let S be an irreducible regular Noetherian prescheme, with field of rational functions Y, let K' be a finite extension of K that is unramified over S, let S' be the normalisation of S in K' (which is thus an étale cover of S), and let X be an abelian scheme over K such that X \otimes _K K' is of the form X'_0 \times _{S'} \operatorname {Spec} (K'), where X'_0 is a projective abelian scheme over S'.
Then X is of the form X_0 \times _S \operatorname {Spec} (K), where X_0 is a projective abelian scheme over S.
621Remarkfga3.i-b.5-remarkfga3.i-b.5-remark.xmlfga3.i-b.5
The speaker does not know if we can replace the hypothesis that S' \to S be a surjective étale cover (which allows us to apply ) with the hypothesis that it is instead a simple and surjective morphism of finite type (not even if we suppose that it is an étalement), or if the proposition still holds true without supposing that X'_0 is projective over S' (a condition which could be automatically satisfied).
623fga3.i-b.6fga3.i-b.6.xmlApplication to local triviality and isotriviality criteriaB.6fga3.i-b
Let S be a prescheme, G a "prescheme of groups" over S, P a prescheme over S on which "G acts" (on the right).
We say that P is formally principal homogeneous for G if the well-known morphism
G \times _S P \to P \times _S P
(induced from the actions of G on P) is an isomorphism.
From now on, we assume G to be flat over S (and thus faithfully flat over S), and we reserve the name of principal homogeneous bundle for G for a formally principal homogeneous bundle P that is faithfully flat and quasi-compact over S.
It is immediate that this is equivalent to being able to find a faithfully flat and quasi-compact extension S' \to S of the base S such that the formally principal homogeneous bundle P'=P \times _S S' for G'=G \times _S S' is trivial, i.e. isomorphic to G' (i.e. admitting a section);
we can take, in particular, S'=P.
Note also that, if S is locally Noetherian, then the faithfully-flat hypothesis on P is equivalent to the hypothesis that \overline {P}_S=P \times _S \operatorname {Spec} ( \overline { \mathscr { O } }_s) be faithfully flat over \overline { \mathscr { O } }_s for all s \in S (where \overline { \mathscr { O } }_s denotes the completion of the local ring \mathscr { O } _s), as follows from the fact that \overline { \mathscr { O } }_s is faithfully flat over \mathscr { O } _s.
Also, if P is of finite type over S, and S is locally Noetherian, then the set of points s satisfying the above condition is constructible, and so, if S is a "Jacobson prescheme" (for example, a scheme of finite type over a field, or, more generally, over a Jacobson ring), then it suffices to verify the condition in question for the closed points of S.
This leads us to the case where the base is the spectrum of a complete local ring A.
If S= \operatorname {Spec} (A) (with A a complete Noetherian local ring), and if P is of finite type over S, then the faithful flatness of P/S is also equivalent to the existence of an S' that is finite and flat over S such that P' is trivial, and, if, further, G is simple over S, then we can suppose S' to be étale over S.
Then, if, further, the residue field of A is algebraically closed (the "geometric case"), then P is faithfully flat over A if and only if it is trivial.
Thus, if S is an algebraic prescheme over an algebraically closed field, and if G is simple and of finite type over S, then we see that the faithfully-flat condition on S is equivalent to the condition of being analytically trivial (SLF) of Serre [Ser1958a, pp.1–12].
We can consider other, stronger, types of conditions on P, that have a "local triviality" nature.
In particular, we say that P is isotrivial (resp. strictly isotrivial) if, for all s \in S, there exists an open neighbourhood U of S, and a finite and faithfully flat morphism (resp. a surjective étale covering) U' \to U such that P'=P \times _S U' is trivial.
(We stray from the terminology of Serre [GD1960], which uses "locally isotrivial" for what we call "strictly isotrivial").
Strict isotriviality is mainly useful if G is simple over S, but is, however, an inadequate notion in other cases.
If G is affine over S, then every principal homogeneous bundle P for G is affine, by , whence the possibility, thanks to , to "descend" from such bundles by faithfully flat and quasi-compact morphisms.
Taking, in particular, G= \operatorname {GL} (n)_S, defined by the condition that the functor S' \mapsto \operatorname {Hom} (S',G) of S-preschemes (with values in the category of groups) can be identified with the functor \operatorname {GL} (n)(S')= \operatorname {GL} (n, \operatorname {H} ^0(S', \mathscr { O } _{S'})) described in .
Using the facts
that every principal homogeneous bundle for G (resp. every locally free sheaf of rank n on S) becomes isomorphic to the "trivial" object G (resp. \mathscr { O } _S^n) under a suitable faithfully flat and quasi-compact extension of S;
that we can descend the type of objects in question (principal homogeneous bundles for G, resp. locally free sheaves of rank n) by such morphisms; and, finally
that the automorphism group of the trivial bundle on any S'/S is functorially isomorphic to the automorphism group of the trivial locally free sheaf of rank n on S',
we "formally" conclude that it is "equivalent" to give, on S (or on some S'/S) a principal homogeneous bundle for the group G, or to give a locally free sheaf of rank n.
(More precisely, we have an equivalence of fibred categories).
We thus conclude, in particular:
298Propositionfga3.i-b.6-proposition-6.1fga3.i-b.6-proposition-6.1.xml6.1fga3.i-b.6
Every principal homogeneous bundle for the group \operatorname {GL} (n)_S is locally trivial.
By known arguments, we thus conclude the same result for others structure groups such as \operatorname {SL} (n)_S, \operatorname {Sp} (n)_S, and products of such groups.
We thus also conclude that, if F is a closed subgroup of G= \operatorname {GL} (n)_S that is flat over S, and such that the quotient G/F exists, and such that G is an isotrivial (resp. strictly isotrivial) principal homogeneous bundle on G/F, of structure group F \times _S(G/F), then every principal homogeneous bundle of structure group F is isotrivial (resp. strictly isotrivial).
This applies to all the "linear groups" on S that have been used up until now, and, in particular, to the case where G=S \times _k \Gamma, with S a prescheme over the field k, and \Gamma a linear group (in the classical sense) over k (and thus in particular simple).
This thus answers, for such groups, a question of Serre (loc. cit.).
We also point out that, for most groups (linear or not) that are simple over S that we know of, and certainly for all those of the form S \times _k \Gamma as above, we can show that every isotrivial principal homogeneous bundle is strictly isotrivial, which answers, in particular, another question of Serre (loc. cit. pp.1–14), taking into account the fact that a homogeneous principal bundle obtained by a descent à la Cartier (cf. ) is obviously isotrivial.
299Remarkfga3.i-b.6-remarkfga3.i-b.6-remark.xmlfga3.i-b.6
One of the essential difficulties in these questions (setting aside the question of the existence of quotient schemes) is the lack of effectiveness criteria for a descent data along a faithfully flat non-finite morphism.
3403indexindex.xmlGrothendieck's "Foundations of Algebraic Geometry" (FGA)
This is an English translation of Alexander Grothendieck's "Fondements de la Géometrie Algébrique".
Any notes by the translator are in italics and prefixed with "[Trans.]".
The original (French) notes have been scanned and uploaded by the Grothendieck Circle here, though you can also find the individual talks, as well as the errata, on Numdam.
The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.
You can view the entire source code of this translation (and contribute or submit corrections) in the GitHub repository.
Corrections and comments welcome.
The translator would like to sincerely thank Steve Hnizdur for catching many typos and mistakes, as well as Jon Sterling for helping with the technical support in getting this translation ported to run on Forester.
[PDF version coming soon]
Foreword
Duality theorems for coherent algebraic sheaves
Formal geometry and algebraic geometry
Technique of descent and existence theorems in algebraic geometry
Generalities, and descent by faithfully flat morphisms
The existence theorem and the formal theory of modules
Quotient preschemes
Hilbert schemes
Picard schemes: Existence theorems
Picard schemes: General properties
Bibliography
3404fga3.iii-introductionfga3.iii-introduction.xmlQuotient preschemes › Introductionfga3.iii1546Remarkfga3.iii-introduction-remarkfga3.iii-introduction-remark.xmlfga3.iii-introduction[Comp.]
We note that the application (of the theory developed here) in ("Picard schemes: Existence theorems") can equally be replaced by a suitable use of Hilbert schemes (cf. Séminaire Mumford–Tate, Harvard University (1961–62)).
As mentioned in , the most important gap in the theory presented here is the lack of an existence criterion for quotients by a non-proper equivalence relation, such as the equivalence relations coming from certain actions of the projective group.
An important theorem in this direction has been obtained by Mumford [@Mum1961].
For a refinement of his result, and various applications the the theory, see Séminaire Mumford–Tate, Harvard University (1961–62).
The problems discussed in the current talk differ from those discussed in the two previous ones, in that we try to represent certain covariant, no longer contravariant, functors of varying schemes.
The procedure of passing to the quotient is, however, essential in many questions of construction in algebraic geometry, including those from and .
Indeed, the question of effectiveness of a descent data on a T-prescheme X, with respect to a faithfully flat and quasi-compact morphism T \to S, is equivalent to the question of existence of a quotient of X (satisfying reasonable properties that we examine below) by the flat equivalence relation on X defined by the descent data;
the questions raised in FGA 3.I, §A.2.c can probably be answered at the same time as the questions posed in of this current talk.
Similarly, the Picard scheme (for the definition, see FGA 3.II, §C.3) of an S-scheme X can be defined in many ways, such as as a quotient of certain other schemes (with positive divisors, or immersions into a projective) by flat equivalence relations, with the definition and construction of these auxiliary schemes being also more simple: they are basically schemes of the type \operatorname {Hom} _S(X,Y), and variants defined in FGA 3.II, §C.2, and their construction will be the subject of the following talk (under suitable hypotheses of projectivity).
Thus, combining the results of the current talk with those of the following, we will obtain the construction of Picard schemes, under suitable hypotheses.
The problem of passing to the quotient in preschemes again offers unresolved questions.
The most important is mentioned in .
It currently remains as the only obstacle to the construction of schemes of modules over the integers for curves of arbitrary degree, polarised abelian varieties, etc.
That is to say, its solution deserves the efforts of specialists of algebraic groups.
3405fga3.iv-introductionfga3.iv-introduction.xmlHilbert schemes › Introductionfga3.iv
The techniques described in and were, for the most part, independent of any projective hypotheses on the schemes in question.
Unfortunately, they have not as of yet allowed us to solve the existence problems posed in .
In the current article, and the following, we will solve these problems by imposing projective hypotheses.
The techniques used are typically projective, and practically make no use of any results from and .
Here we will construct "Hilbert schemes", which are meant to replace the use of Chow coordinates, as was mentioned in FGA 3.II, §C.2.
In the next article, the theory of passing to the quotient in schemes, developed in , combined with the theory of Hilbert schemes, will allow us, for example, to construct Picard schemes (defined in FGA 3.II, §C.3) under rather general conditions.
In summary, we can say that we now have a more or less satisfying technique of projective constructions, apart from the fact that we are still missing a theory of passing to the quotient by groups such as the projective group, acting "without fixed points" (cf. FGA 3.III, §8).
The situation even seems slightly better in analytic geometry (if we restrict to the study of "projective" analytic spaces over a given analytic space), since, for analytic spaces, the difficulty of passing to the quotient by a group that acts nicely disappears.
Either way, in algebraic geometry, as well as in analytic geometry, it remains to develop a construction technique that works without any projective hypotheses.
3406fga3.ii-afga3.ii-a.xmlThe existence theorem and the formal theory of modules › Representable and pro-representable functorsAfga3.ii2460fga3.ii-a.1fga3.ii-a.1.xmlRepresentable functorsA.1fga3.ii-a
Let \mathcal {C} be a category.
For all X \in \mathcal {C}, let h_X be the contravariant functor from \mathcal {C} to the category \mathtt {Set} of sets given by
\begin {aligned} h_X \colon \mathcal {C} & \to \mathtt {Set} \\ Y& \mapsto \operatorname {Hom} (Y,X). \end {aligned}
If we have a morphism X \to X' in \mathcal {C}, then this evidently induces a morphism h_X \to h_{X'} of functors;
h_X is a covariant functor in X, i.e. we have defined a canonical covariant functor
h \colon \mathcal {C} \to \operatorname {Hom} ( \mathcal {C}^ \circ , \mathtt {Set} )
from \mathcal {C} to the category of covariant functors from the dual \mathcal {C}^ \circ of \mathcal {C} to the category of sets.
We then recall:
807Propositionfga3.ii-a.1-proposition-1.1fga3.ii-a.1-proposition-1.1.xml1.1fga3.ii-a.1
This functor h is fully faithful;
in other words, for every pair X,X' of objects of \mathcal {C}, the natural map
\operatorname {Hom} (X,X') \to \operatorname {Hom} (h_X,h_{X'})
is bijective.
In particular, if a functor F \in \operatorname {Hom} ( \mathcal {C}^ \circ , \mathtt {Set} ) is isomorphic to a functor of the form h_X, then X is determined up to unique isomorphism.
We then say that the functor F is representable.
The above proposition then implies that the canonical functor h defines an equivalence between the category \mathcal {C} and the full subcategory of \operatorname {Hom} ( \mathcal {C}^ \circ , \mathtt {Set} ) consisting of representable functors.
This fact is the basis of the idea of a "solution of a universal problem", with such a problem always consisting of examining if a given (contravariant, as here, or covariant, in the dual case) functor from \mathcal {C} to \mathtt {Set} is representable.
Note further that, just by the definition of products in a category [Gro1957], the functor h \colon X \mapsto h_X commutes with products whenever they exist (and, more generally, with finite or infinite projective limits, and, in particular, with fibred products, taking "kernels" [], etc., whenever such things exist): we have an isomorphism of functors
h_{X \times X'} \xrightarrow { \sim } h_X \times h_{X'}
whenever X \times X' exists, i.e. we have functorial (in Y) bijections
h_{X \times X'} \xrightarrow { \sim } h_X(Y) \times h_{X'}(Y).
In particular, the data of a morphism
X \times X' \to X''
in \mathcal {C} (i.e. of a "composition law" in \mathcal {C} between X, X', and X'') is equivalent to the data of a morphism h_{X \times X'}=h_X \times h_{X'} \to h_{X''}, i.e. to the data, for all Y \in \mathcal {C}, of a composition law of sets
h_X(Y) \times h_{X'}(Y) \to h_{X''}(Y)
such that, for every morphism Y \to Y' in \mathcal {C}, the system of set maps
h_{X^{(i)}}(Y) \to h_{X^{(i)}}(Y') \qquad \text {for }i=0,1,2
is a morphism for the two composition laws, with respect to Y and Y'.
In this way, we see that the idea of a "\mathcal {C}-group" structure, or a "\mathcal {C}-ring" structure, etc. on an object X of \mathcal {C} can be expressed in the most manageable way (in theory as much as in practice) by saying that, for every Y \in \mathcal {C}, we have a group law (resp. ring law, etc.) in the usual sense on the set h_X(Y), with the maps h_X(Y) \to h_X(Y') corresponding to morphisms Y \to Y' that should be group homomorphisms (resp. ring homomorphisms, etc.).
This is the most intuitive and manageable way of defining, for example, the various classical groups \operatorname {G_a}, \operatorname {G_m}, \operatorname {GL} (n), etc. on a prescheme S over an arbitrary base, and of writing the classical relations between these groups, or of placing a "vector bundle" structure on the affine scheme V( \mathscr { F } ) over S defined by a quasi-coherent sheaf \mathscr { F }, and of defining and studying the many associated flag varieties (Grassmannians, projective bundles), etc.;
the general yoga is purely and simply identifying, using the canonical functor h, the objects of \mathcal {C} with particular contravariant functors (namely, representable functors) from \mathcal {C} to the category of sets.
The usual procedure of reversing the arrows that is necessary, for example, in the case of affine schemes in order to pass from the geometric language to the language of commutative algebra, leads us to dualise the above considerations, and, in particular, to also introduce covariant representable functors \mathcal {C} \to \mathtt {Set}, i.e. those of the form Y \mapsto \operatorname {Hom} (X,Y)=h'_X(Y).
2461fga3.ii-a.2fga3.ii-a.2.xmlPro-representable functors, pro-objectsA.2fga3.ii-a
Let \mathcal {X}=(X_i)_{i \in I} be a projective system of objects of \mathcal {C};
there is a corresponding covariant functor
h'_{ \mathcal {X}} = \varinjlim _i h'_{X_i}
which can be written more explicitly as
h'_{ \mathcal {X}}(Y) = \varinjlim _i h'_{X_i}(Y) = \varinjlim _i \operatorname {Hom} (X_i,Y)
which is a functor from \mathcal {C} to \mathtt {Set}.
A functor from \mathcal {C} to \mathtt {Set} that is isomorphic to a functor of this type with I filtered is said to be pro-representable.
By the previous section, these are exactly the functors that are isomorphic to filtered inductive limits of representable functors.
Let \mathcal {X}'=(X_j)_{j \in J} be another filtered projective system in \mathcal {C} (indexed by another filtered preordered set of indices J).
Then we can easily show that we have a canonical bijection
\operatorname {Hom} (h_{ \mathcal {X}'},h_{ \mathcal {X}}) = \varprojlim _j \varinjlim _i \operatorname {Hom} (X_i,X'_j)
(generalising ).
This leads to introducing the category \operatorname {Pro} ( \mathcal {C}) of pro-objects of \mathcal {C}, whose objects are projective systems of objects of \mathcal {C} (indexed by arbitrary filtered preordered sets of indices), and whose morphisms between objects \mathcal {X}=(X_i)_{i \in I} and \mathcal {X}'=(X_j)_{j \in J} are given by
\operatorname {Pro} \operatorname {Hom} ( \mathcal {X}, \mathcal {X}') = \varprojlim _j \varinjlim _i \operatorname {Hom} (X_i,X'_j),
with the composition of pro-homomorphisms being evident.
By the very construction itself, \mathcal {X} \mapsto h'_{ \mathcal {X}} can be considered as a contravariant functor in \mathcal {X}, establishing an equivalence between the dual category of the category \operatorname {Pro} ( \mathcal {C}) of pro-objects of \mathcal {C} and the category of pro-representable covariant functors from \mathcal {C} to \mathtt {Set}.
Of course, an object X of \mathcal {C} canonically defines a pro-object, denoted again by X, so that \mathcal {C} is equivalent to a full subcategory of \operatorname {Pro} ( \mathcal {C}).
Then, if \mathcal {X}=(X_i)_{i \in I} is an arbitrary pro-object of \mathcal {C}, then (with the above identification) we have that
\mathcal {X} = \varprojlim _i X_i
with the projective limit being taken in \operatorname {Pro} ( \mathcal {C}) (since h_{ \mathcal {X}}= \varinjlim _i h_{X_i}).
We draw attention to the fact that, even if the projective limit of the X_i exists in \mathcal {C}, it will generally not be isomorphic to the projective limit \mathcal {X} in \operatorname {Pro} ( \mathcal {C}), as is already evident in the case where \mathcal {C} is the category of sets.
We note that, by the definition itself, \varprojlim {}_{ \mathcal {C}}X_i=L is defined by the condition that the functor
\varprojlim _i \operatorname {Hom} _{ \mathcal {C}}(Y,X_i) = \operatorname {Hom} _{ \operatorname {Pro} ( \mathcal {C})}(Y, \mathcal {X})
in Y \in \mathcal {C} and with values in \mathtt {Set} be representable via \mathcal {L}, i.e. that it be isomorphic to \operatorname {Hom} _{ \mathcal {C}}(Y, \mathcal {L});
then \lim {}_{ \mathcal {C}}X_i is already defined in terms of the pro-object \mathcal {X}, and, in a precise way, depends functorially on the pro-object \mathcal {X} whenever it is defined;
there is therefore no problem with denoting it by \lim {}_{ \mathcal {C}}( \mathcal {X}).
If projective limits in \mathcal {C} always exist, then \lim {}_{ \mathcal {C}}( \mathcal {X}) is a functor from \operatorname {Pro} ( \mathcal {C}) to \mathcal {C}, and there is a canonical homomorphism of functors \lim _ \mathcal {C}( \mathcal {X}) \to \mathcal {X}.
Since every (covariant, say, for simplicity) functor
F \colon \mathcal {C} \to \mathcal {C}'
can be extended in an obvious way to a functor
\operatorname {Pro} (F) \colon \operatorname {Pro} ( \mathcal {C}) \to \operatorname {Pro} ( \mathcal {C}'),
it follows that, if projective limits always exist in \mathcal {C}', then F also canonically defines a composite functor
\overline {F} = \varprojlim {}_{ \mathcal {C}'} \colon \operatorname {Pro} ( \mathcal {C}) \to \mathcal {C}'
sending \mathcal {X}=(X_i)_{i \in I} to \varprojlim {}_{ \mathcal {C}'}F(X_i).
A pro-object \mathcal {X} is said to be a strict pro-object if it is isomorphic to a pro-object (X_i)_{i \in I}, where the transition morphisms X_i \to X_j are epimorphisms;
a functor defined by such an object is said to be strictly pro-representable.
We can thus further demand that I be a filtered ordered set, and that every epimorphism X_i \to X' be equivalent to an epimorphism X_i \to X_j for some suitable j \in I (uniquely determined by this condition).
Under these conditions, the projective system (X_i)_{i \in I} is determined up to unique isomorphism (in the usual sense of isomorphisms of projective systems).
It thus follows that the projective limit of a projective system \mathcal {X}^{( \alpha )} of strict pro-objects always exists in \operatorname {Pro} ( \mathcal {C}), and that, with the above notation of F and \overline {F}, we have that
\overline {F} \varprojlim _ \alpha \mathcal {X}^{( \alpha )} = \varprojlim _ \alpha {}_{ \mathcal {C}'}F(X^{( \alpha )}).
In particular, if every pro-object of \mathcal {C} is strict (cf. the previous section), then the extended functor \overline {F} commutes with projective limits.
2462fga3.ii-a.3fga3.ii-a.3.xmlCharacterisation of pro-representable functorsA.3fga3.ii-a
Let \mathcal {C} and \mathcal {C}' be categories in which all finite projective limits (i.e. limits over finite, not necessarily filtered, preordered sets) exist, or, equivalently, in which finite products and finite fibred products exist (which implies, in particular, the exists of a "right-unit object" e such that \operatorname {Hom} (X,e) consists of only on element for all X).
Let F be a covariant functor from \mathcal {C} to \mathcal {C}'.
Then the following conditions are equivalent:
F commutes with finite projective limits;
F commutes with finite products and with finite fibred products;
F commutes with finite products, and, for every exact diagram
X \to X' \rightrightarrows X''
in \mathcal {C} (cf. FGA 3.I, A, Definition 2.1), the image of the diagram under F
F(X) \to F(X') \rightrightarrows F(X'')
is exact.
We then say that F is left exact.
In what follows, we assume that finite projective limits always exist in \mathcal {C}.
It is then immediate from the definitions that a representable functor is left exact, and, by taking the limit, that a pro-representable functor is left exact.
To obtain a converse, let
F \colon \mathcal {C} \to \mathtt {Set}
be a covariant functor, and let X \in \mathcal {C} and \xi \in F(X).
We say that \xi (or the pair (X, \xi )) is minimal if, for all X' \in \mathcal {C} and all \xi ' \in F(X'), and for every strict monomorphism (cf. FGA 3.I, §A.2) u \colon X' \to X such that \xi =F(u)( \xi '), u is an isomorphism.
We also say that a pair (X, \xi ) dominates (X'', \xi '') (where \xi \in F(X) and \xi '' \in F(X'')) if there exists a morphism v \colon X \to X'' such that \xi ''=F(v)( \xi );
if \xi is minimal, and if F is left exact, then this morphism v is unique;
if \xi '' is minimal, then v is surjective.
From this we easily deduce the following proposition:
1505Propositionfga3.ii-a.3-proposition-3.1fga3.ii-a.3-proposition-3.1.xml3.1fga3.ii-a.3
For F to be strictly pro-representable, it is necessary and sufficient that it satisfy the following two conditions:
F is left exact; and
every pair (X, \xi ), with \xi \in F(X), is dominated by some minimal pair.
This second condition is trivial if every object of \mathcal {C} is Artinian (by taking a sub-object X' of X that is minimal amongst those for which there exists some \xi ' \in F(X') such that \xi is the image of \xi ').
Whence:
1506Corollaryfga3.ii-a.3-proposition-3.1-corollaryfga3.ii-a.3-proposition-3.1-corollary.xmlfga3.ii-a.3
Let \mathcal {C} be a category whose objects are all Artinian and in which all finite projective limits exist.
Then the pro-representable functors from \mathcal {C} to \mathtt {Set} are exactly the left exact functors, and they are in fact strictly pro-representable.
This last fact also implies that every pro-object of \mathcal {C} is then strict.
2463fga3.ii-a.4fga3.ii-a.4.xmlExample: groups of Galois type, pro-algebraic groupsA.4fga3.ii-a
If \mathcal {C} is the category of ordinary finite groups, then \operatorname {Pro} ( \mathcal {C}) is equivalent to the category of totally disconnected compact topological groups.
([Trans.] Here the word "Hausdorff" is implicit.)
It is groups of this type, and their generalisations, obtained by replacing ordinary finite groups with schemes of finite groups over a given base prescheme (for example, finite algebraic groups over a field k), that serve as fundamental groups, homotopy groups, and absolute and relative homology groups for preschemes.
In all these examples, the corollary to applies, and it is indeed by the associated functor that the \pi _1 should be defined [].
It is the same if we start with the category of algebraic or quasi-algebraic groups over a field (or, more generally, over a Noetherian prescheme): we recover the "pro-algebraic groups" of Serre [Ser1958].
2464fga3.ii-a.5fga3.ii-a.5.xmlExample: "formal varieties"A.5fga3.ii-a
Let \Lambda be a Noetherian ring, \mathcal {C} the category of \Lambda-algebras that are Artinian modules of finite type over \Lambda (or, more concisely, Artinian \Lambda-algebras).
The conditions of the corollary to
are then satisfied.
Here, the category \operatorname {Pro} ( \mathcal {C}) is equivalent to the category of topological algebras O over \Lambda that are isomorphic to topological projective limits
O = \varprojlim O_i
of algebras O_i \in \mathcal {C}, i.e. those whose topology is linear, separated, and complete, and such that, for every open ideal \mathfrak {J}_i of O, the algebra O/ \mathfrak {J}_i is an Artinian algebra over \Lambda.
The functor \mathcal {C} \to \mathtt {Set} associated to such an algebra is exactly
\begin {aligned} F(A) &= h'_{O}(A) \\ &= \{ \text {continuous homomorphisms of topological } \Lambda \text {-algebras }O \to A \} \\ &= \varinjlim _i \operatorname {Hom} _{ \Lambda \text {-algebras}}(O_i,A). \end {aligned}
Note also that the category \mathcal {C} is essentially the product of analogous categories, corresponding to the local rings that are the completions of the \Lambda _{ \mathfrak {m}} for the maximal ideals \mathfrak {m} of \Lambda;
we can thus, if so desired, restrict to the case where A is such a complete local ring.
In any case, O decomposes canonically as the topological product of its local components, which correspond to the "points" of the formal scheme [] defined by O.
Such a point is defined by an object \xi of some F(K), where K \in \mathcal {C} is a field (for example, the residue field of the local component in question), and where two pairs ( \xi ,K) and ( \xi ',K') define the same point if and only if they are both dominated by the same ( \xi '',K''), or if they both dominate the same ( \xi ''',K''').
(If the { \Lambda }/ \mathfrak {m} are algebraically closed, then it suffices to take the set given by the sum of the F({ \Lambda }/ \mathfrak {m})).
It is important to give conditions that ensure that the local component O_ \xi of O corresponding to some \xi \in F(K) be a Noetherian ring.
If \Lambda is a complete local ring (Noetherian, we recall), then it is equivalent to say that O_ \xi is isomorphic to a quotient ring of a formal series ring \Lambda [{[t_1, \ldots ,t_n]}].
To give such a criterion, we introduce (for every ring A) the A-algebra I_A of "dual numbers" of A, defined by
I_A = A[t]/t^2A[t].
Let \varepsilon \colon I_A \to A be the augmentation homomorphism, which defines (if A \in \mathcal {C}) a map
F( \varepsilon ) \colon F(I_A) \to F(A).
Using the fact that F is left exact, we intrinsically define the structure of an A-module on the subset
F(I_A, \xi ) = F( \xi )^{-1}( \xi ) \subset F(I_A)
consisting of the \xi ' \in F(I_A) that are "reducible along \xi";
using the explicit form of F in terms of O, we find that this K-module can be identified with \operatorname {Hom} _ \Lambda ( \mathfrak {m}_{ \xi }/ \mathfrak {m}_ \xi ^2,A), where m_ \xi is the kernel of the homomorphism \xi \colon O \to A, i.e. if A is a field, then the maximal ideal of the local component O_ \xi of O.
From this, we immediately deduce the following proposition:
524Propositionfga3.ii-a.5-proposition-5.1fga3.ii-a.5-proposition-5.1.xml5.1fga3.ii-a.5
Let \xi \in F(K), where K \in \mathcal {C} is a field.
For the corresponding local component O_ \xi of O to be a Noetherian ring, it is necessary and sufficient that the set F(I_K, \xi ) of elements of F(I_K) that are reducible along \xi be a vector space of finite dimension over K.
Under these conditions, we have a canonical isomorphism
F(I_K, \xi ) = \operatorname {Hom} ( \mathfrak {m}_{ \xi }/ \mathfrak {m}_ \xi ^2+ \mathfrak {n}_ \xi \mathscr { O } _ \xi , K)
(where \mathfrak {n}_ \xi is the maximal ideal of \Lambda given by the kernel of the homomorphism \Lambda \to K), and so, in particular, the dimension of the K-vector space F(I_K, \xi ) is equal to the dimension of the vector space \mathfrak {m}_{ \xi }/ \mathfrak {m}_ \xi ^2 over the field O_{ \xi }/ \mathfrak {m}_ \xi =K( \xi ).
[Comp.]
The formula given above is only correct when \Lambda is a field; in the general case, we must replace \mathfrak {m}_{ \xi }/ \mathfrak {m}_ \xi ^2 with the quotient of this space by the image of \mathfrak {n}_{ \xi }/ \mathfrak {n}_ \xi ^2, where \mathfrak {n} is the maximal ideal of \Lambda.
Suppose that O_ \xi is Noetherian, and suppose, for notational simplicity, that \Lambda is complete and local, and that O=O_ \xi.
([Comp.] The following definition is correct only when the residue extension k'/k is separable; for the general case, see [Gro1960b, III, 1.1].)
We say that O is simple over \Lambda if O is a finite and étale algebra over the completion algebra of the localisation of \Lambda [t_1, \ldots ,t_n] at one of its maximal ideals that induces the maximal ideal of \Lambda;
if the residue extension of O over \Lambda is trivial (for example, if the residue field of \Lambda is algebraically closed), then this is equivalent to saying that O itself is isomorphic to such a formal series algebra.
Finally, if we no longer necessarily suppose that O is Noetherian, then we again say that O is simple over \Lambda if O is isomorphic to a topological projective limit of quotient \Lambda-algebras that are Noetherian and \Lambda-simple in the above sense.
We can immediately generalise to the case where \Lambda and O are no longer assumed to be local.
With this, we have the following proposition:
525Propositionfga3.ii-a.5-proposition-5.2fga3.ii-a.5-proposition-5.2.xml5.2fga3.ii-a.5
For O to be simple over \Lambda, it is necessary and sufficient that the associated functor F send epimorphisms to epimorphisms.
If this is the case, then this implies that, for every surjective homomorphism A \to A' in \mathcal {C}, the morphism F(A) \to F(A') is also surjective.
Of course, it suffices to verify this condition in the case where A is local, and (proceeding step-by-step) where the ideal of A given by the kernel of A \to A' is annihilated by the maximal ideal of A.
This leads, in practice, to verifying that a certain obstruction, linked to infinitesimal invariants of the situation that give us a functor F, is null;
this is a problem of a cohomological nature.
To finish, we say some words, in the above context, about rings of definition.
Let F still be a functor from \mathcal {C} to \mathtt {Set}, assumed to be pro-representable via a topological \Lambda-algebra O.
Then, for every A \in \mathcal {C} and every \xi \in F(A), there exists a smallest subring A' of A such that \xi is the image of an element \xi ' of F(A') (which is then uniquely determined):
indeed, it suffices to think of \xi as a homomorphism from O to A, and to take A' to be the image of O under this \xi.
We then say that A' is the ring of definition of the object \xi \in F(A).
If u \colon A \to B is an algebra homomorphism, and if \eta =F(u)( \xi ), then the ring of definition of \eta is the image under u of the ring of definition of \xi.
If we start with a functor F from \mathcal {C} to \mathtt {Set}, then the existence of rings of definition, along with their properties that we have just discussed, is more or less equivalent to the condition that F be pro-representable;
that is, they are usually far from being trivial.
3407fga3.iv-7fga3.iv-7.xmlHilbert schemes › Supplements and questions7fga3.iv
As remarked by J.-P. Serre, it follows from a well-known example of Nagata that we can find a scheme S that is the spectrum of a field k, an S-scheme S' that is the spectrum of a quadratic extension k' of k, and finally a simple and proper (but non-projective) S'-scheme X of dimension 3 such that \prod _{S'/S}(X/S) does not exist.
This implies a fortiori that the Hilbert scheme \underline { \operatorname {Hilb} } _{X/S}^2 does not exist (nor even the k-scheme that would represent the étale covers of rank 2 of S contained inside X, nor a fortiori the symmetric square of X, cf. ).
This thus imposes serious limitations on the possibilities of non-projective constructions in algebraic geometry.
(It is, however, plausible that such limitations do not present themselves in analytic geometry, just as they do not present themselves in formal geometry (cf. )).
However, if X is a proper scheme over the spectrum S of a field k, and if Z is quasi-projective over X, then \prod _{X/S}(Z/X) exists, and is a scheme, given by the sum of a sequence of quasi-projective schemes over S (as in the projective case ).
To see this, we can reduce to the case where X is itself projective, by dominating X by a projective S-scheme X';
we will not give here the details of the proof, which also uses the result of factorisation of a finite morphism given in FGA 3.I, §A.2.b.
The success of the method is all in the fact that, with S the spectrum of a field, the X' that appears in Chow's lemma will automatically be flat over S.
I do not know if the result remains true without any hypotheses on S, supposing only that X is proper and flat over S, and that Z is quasi-projective over X.
An important case in the applications is that where Z is a closed subscheme of X;
if then \prod _{X/S}(Z/X) exists, it is necessarily a closed subscheme of S.
We can construct it directly in a relatively simple manner whenever X is projective over S, without using the theory of Hilbert schemes, and the method used shows more generally that, if Z is affine over X, then \prod _{X/S}(Z/X) exists and is affine over S.
It equally shows that, if X is proper and flat over S (but not necessarily projective over S), then, for every vector bundle Z that is locally trivial on X, \prod _{X/S}(Z/X) exists and is a vector bundle on S.
It would be desirable for these results to be studied again and unified.
3408fga3.ii-bfga3.ii-b.xmlThe existence theorem and the formal theory of modules › The two existence theoremsBfga3.ii
Keeping the notation of , and, given a covariant functor
F \colon \mathcal {C} \to \mathtt {Set} ,
we wish to find manageable criteria for F to be pro-representable, i.e. expressible via a \Lambda-algebra O as above.
By the corollary of §A, Proposition 3.1, to ensure this, it is necessary and sufficient that F be left exact.
In the current state of the technique of descent (cf. the questions asked in FGA 3.I, §A.2.c), this criterion is not directly verifiable, in this form, in the most important cases, and we need criteria that seem less demanding.
1549Theoremfga3.ii-b-theorem-1fga3.ii-b-theorem-1.xml1fga3.ii-b
For the functor F to be pro-representable, it is necessary and sufficient that it satisfy the two following conditions:
F commutes with finite products;
for every algebra A \in \mathcal {C} and every homomorphism A \to A' in \mathcal {C} such that the diagram
A \to A' \rightrightarrows A' \otimes _A A'
is exact (cf. FGA 3.I, §A, Definition 1.2), the diagram
F(A) \to F(A') \rightrightarrows F(A' \otimes _A A')
is also exact.
Furthermore, it suffices to verify condition (ii) in the case where A is local, and when, further, we are in one of the two following cases:
A is a free module over A;
the quotient module A'/A is an A-module of length 1.
1441Prooffga3.ii-b-theorem-1
The proof of this theorem is rather delicate, and cannot be sketched here.
We content ourselves with pointing out that it relies essentially on a study of equivalence relations (in the sense of categories) in the spectrum of an Artinian algebra (the study of which poses even more problems, whose solutions seems essential for the further development of the theory).
In applications, the verification of condition (i) is always trivial.
The verification of condition (ii) splits into two cases: case (a), where A' is a free A-module, can be dealt with using the technique of descent by flat morphisms (cf. FGA 1, Theorems 1, 2, and 3), which offers no difficulty;
to deal with case (b), we will use the following result:
1550Theoremfga3.ii-b-theorem-2fga3.ii-b-theorem-2.xml2fga3.ii-b
Let A be a local Artinian ring with maximal ideal \mathfrak {m}, and let A' be an A-algebra containing A, and such that \mathfrak {m}A' \subset A, and A \to A' \rightrightarrows A' \otimes _A A' is exact (which is the case, in particular, if A'/A is an A-module of length 1).
Let \mathcal {F} be the fibred category (cf. FGA 3.I, §A, Definition 1.1) of quasi-coherent sheaves that are flat over varying preschemes.
Then the morphism \operatorname {Spec} (A') \to \operatorname {Spec} (A) is a strict \mathcal {F}-descent morphism (cf. FGA 3.I, §A, Definition 1.7).
1428Prooffga3.ii-b-theorem-2
We prove this by first proving that
\operatorname {H} ^i(A'/A, \operatorname {G_a} ) = 0 \qquad \text {for }i \geqslant1
(cf. FGA 3.I, §A.4.e), with the hypothesis that \mathfrak {m}A' \subset A allowing us to easily reduce to the case where A is a field (namely A/ \mathfrak {m}).
We can then apply the equivalences described in FGA 3.I, §A.4.e.
In other words, the data of a flat A-module M is completely equivalent to the data of a flat A'-module M' endowed with an (A' \otimes _A A')-isomorphism from M' \otimes _A A' to A' \otimes _A M' satisfying the usual transitivity condition for a descent data (loc. cit.).
3393ReferenceSer1958Ser1958.xmlCorps locaux et isogénies1960J.P. SerreSéminaire Bourbaki 11 Talk no. 185@article{Ser1958,
title = {Corps locaux et isog\'{e}nies},
author = {Serre, J.-P.},
year = {1960},
journal = {S\'{e}minaire Bourbaki},
volume = {11},
pages = {Talk no. 185},
}3395ReferenceGro1960bGro1960b.xmlSéminaire de Géométrie Algèbrique1960A. GrothendieckParis, Institut des Hautes Études Scientifiques@book{Gro1960b,
title = {{{S\'{e}minaire de G\'{e}om\'{e}trie Alg\'{e}brique}}},
author = {Grothendieck, A.},
year = {1960/61},
publisher = {{Paris, Institut des Hautes \'{E}tudes Scientifiques}}
}3397ReferenceGro1957Gro1957.xmlSur quelques points d'algèbre homologique1957A. GrothendieckTohoku math. J. 9 pp. 119–221@article{Gro1957,
title = {Sur quelques points d'alg\`{e}bre homologique},
author = {Grothendieck, A.},
year = {1957},
journal = {Tohoku math. J.},
volume = {9},
pages = {119--221},
}