On coherent algebraic and analytic sheaves
4th and 11th of February, 1957
Translator’s note
This page is a translation into English of the following:
Grothendieck, A. “Sur les faisceaux algébriques et les faisceaux analytiques cohérents.” Séminaire Henri Cartan 9 (1956–1957), Talk no. 2. numdam.org/item/SHC_1956-1957__9__A2_0/
The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.
Version: 94f6dce
The aim of this talk is to generalise certain theorems of Serre. It makes fundamental use of the techniques of Serre [1–3].
1 Generalities on coherent algebraic sheaves
Let
Let
—
{\mathscr{O}}_X is a coherent sheaf of rings.- If
X is affine with coordinate ringA(X) , then, for every coherent{\mathscr{O}} -module{\mathscr{A}} onX , the stalks{\mathscr{A}}_x are generated by the canonical image of\Gamma(X,{\mathscr{A}}) . Furthermore,\Gamma(X,{\mathscr{A}}) is anA(X) -module of finite type, and everyA(X) -module of finite type comes from an essentially unique coherent{\mathscr{O}} -module. (Recall that\Gamma(X,{\mathscr{A}}) denotes the module of sections of{\mathscr{A}} overX ). - Under the conditions of b), we have that
\mathrm{H}^i(X,{\mathscr{A}})=0 fori>0 .
Proof. For the proofs, which are very elementary, see [1, chapitre 2, paragraphes 2,3,4], or a talk of Cartier in the 1957 Séminaire Grothendieck.
2 A dévissage theorem
Let
Let
Proof. The proof is done by induction on
Let
Proof. By “compactness” reasons, we can restrict to the case where
Under the above conditions,
This implies that
Suppose first of all that
If
Proof. We can immediately restrict to the case where
Using the exact sequence
Let
On any irreducible algebraic set
Proof. This is an easy consequence of the fact that every open subset of
We will thus identify
Now, if
We say that the subcategory
3 Complements on sheaf cohomology
Let
If
Now let
From the Leray spectral sequence, we get homomorphisms
For the results of this section, see the 1957 Séminaire Grothendieck.
4 Supplementary results on algebraic sheaves on projective space
Let
—
- Let
Y be an affine algebraic set, and{\mathscr{A}} a coherent algebraic sheaf on\mathbf{P}\times Y . Then, for everyn large enough,{\mathscr{A}}(n) is generated by the module of its sections, i.e.{\mathscr{A}}(n) is isomorphic to some quotient of{\mathscr{O}}_{\mathbf{P}\times Y}^k , for some integerk . - For
n large enough,\mathrm{H}^i(\mathbf{P},{\mathscr{O}}(n))=0 .
Proof. The proof is elementary;
for (a), see [1, théorème 1] (where the proof is given for the case where
Now suppose that
—
- Let
{\mathscr{A}}^h be a coherent{\mathscr{O}}^h -module on\mathbf{P}^h . Then, for alln large enough,{\mathscr{A}}^h(n) is isomorphic to a quotient of({\mathscr{O}}^h)^k , for some integerk . - For
n large enough,\mathrm{H}^i(\mathbf{P}^h,{\mathscr{O}}^h(n))=0 .
Proof. The proof is distinctly deeper: see [2, lemme 5, page 12, and lemma 8, page 24].
It works by induction on the dimension, and makes essential use of the fact that the cohomology
5 The finiteness theorem: statement
Let
A morphism
A more geometric definition is the following:
Let
Proof. The proof will be given in §7.
We state here the following corollary, obtained by taking
Let
6 An algebraic-analytic comparison theorem: statement
Let
We will see that, if
This functorial homomorphism can be extended, in a unique way, to functorial homomorphisms (that commute with the coboundary operators):
These homomorphisms have all the functorial properties that we might desire, but whose precise statements will not be given here (even though they will, of course, be essential in the proofs.)
Suppose that the morphism of algebraic sets
Proof. The proof will be given in the following section.
Taking
If
Since
Under the conditions of Theorem 5, the
It is very plausible that, more generally, if
Under the conditions of Theorem 5, suppose further that
Proof. We have already said that
7 Proof of Theorems 4 and 5
The proofs follow mainly from Theorem 3, the “dévissage” of Theorem 2 (which is necessary since there is no reason for
(Chow’s lemma.) —
Let
Recall (§5) that "proper implies, in this case, that the graph of
Proof. We cover
Theorem 4 and Theorem 5 say that every coherent algebraic sheaf
Let
First we will show how this lemma will imply the previous one.
Applying the lemma to
It thus remains only to prove Lemma 5.
Since the graph
We first prove Lemma 5 in the case where
To prove Lemma 5 in the general case, we proceed by induction on
The last paragraph of this proof can be simplified if we use the fact that
8 Algebraic and analytic sheaves on a compact algebraic variety
We are going to complete Corollary 1 of Theorem 5:
Let
The uniqueness of
With
Proof. This homomorphism comes from, by taking sections, the monomorphism of sheaves
From Corollary 1 and the exactness of the functor
Let
Proof. (Proof of Theorem 6.) —
We can now prove Theorem 6, by induction on