3020Volumehodge-theory-iiihodge-theory-iii.xmlHodge Theory IIIIII
P. Deligne.
"Théorie de Hodge II".
Pub. Math. de l'IHÉS 44 (1974) pp. 5–77.
publications.ias.edu/node/3612766hodge-theory-iii-introductionhodge-theory-iii-introduction.xmlIntroductionhodge-theory-iii
This article follows Hodge Theory I and Hodge Theory II;
the numbering of the sections here follows that of (which contains to ).
In , we gave an introduction to the yoga that underlies and the present article .
However, and are logically independent of ;
they also do not contain all the results that were stated in .
In , we introduced the Hodge theory of non-singular (not necessarily complete) algebraic varieties.
Here we treat the case of arbitrary singularities.
From [hodge-theory-iii-7 (?)] onwards, we will make essential use of results from to .
In the case of a complete singular algebraic variety X, the fundamental idea is to use the resolution of singularities to replace X by a simplicial system of smooth projective schemes
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
X_\bullet = \Big (\ldots \quad
X_2
\ar [r]
\ar [r,shift left=4]
\ar [r,shift right=4]
& X_1
\ar [r,shift left=2]
\ar [r,shift right=2]
\ar [l,shift right=2]
\ar [l,shift left=2]
& X_0\Big )
\ar [l]
\end {tikzcd}
(or, as we will say, by a smooth projective simplicial scheme).
The results of [hodge-theory-iii-5 (?)] and [hodge-theory-iii-6 (?)] allow us to find such simplicial schemes X_\bullet that have, in a suitable sense, the same cohomology as X.
We can then "express", via a spectral sequence, the cohomology of X in terms of the cohomology of the X_n.
Following the general principles of , we can then, on each X_n, use classical Hodge theory, and obtain an induced mixed Hodge structure on the cohomology of X.
In the case of an arbitrary algebraic variety X, we "replace" X by a smooth simplicial scheme X_\bullet with compactification given by a smooth projective simplicial scheme \bar {X}_\bullet .
We can further arrange things so that \bar {X}_n\setminus X_n is a normal crossing divisor, given by a union of smooth divisors D_{n,i}.
We thus "express", via a spectral sequence, the cohomology of X in terms of the cohomology of the p-fold intersections of the D_{n,i} (for n,p\geqslant 0), and we obtain an induced mixed Hodge structure on H^\bullet (X).
[hodge-theory-iii-5 (?)] and [hodge-theory-iii-6 (?)] constitute a summary (without proofs) of the theory of "cohomological descent".
This theory, in the more general framework of a topos, is laid out in [SD].
In [hodge-theory-iii-7.1 (?)], we restate certain results obtained previously, but in the language of filtered derived categories;
[hodge-theory-iii-7.2 (?)] contains a simpler proof of the two-filtrations lemma than that of .
The unappealing [hodge-theory-iii-8.1 (?)] is the heart of this work.
The essential result is [hodge-theory-iii-8.1.15 (?)].
In its proof, to be able to apply the two-filtrations lemma, we use the fact that every morphism of mixed Hodge structures is strictly compatible with the filtrations W and W ().
In [hodge-theory-iii-8.2 (?)], we define the mixed Hodge structure of \operatorname {H}^n(X,\mathbb {Z}) (for X a separated scheme).
We show that the Hodge numbers h^{pq} of \operatorname {H}^n(X,\mathbb {Z}) can only be non-zero for (p,q)\in [0,n]\times [0,n].
For n>\dim X, we have a more precise result ([hodge-theory-iii-8.2.4 (?)]).
If X is smooth (resp. complete), they can only be non-zero if, further, p+q\geqslant n (resp. p+q\leqslant n).
If X is a complete subscheme of Z, with complement U, with Z complete and smooth of dimension n, then we can, by N. Katz, interpret this "duality" between the complete case and the smooth case as coming from the "Alexander duality"
\to \ldots \to \operatorname {H}^i(Z,\mathbb {Q}) \to \operatorname {H}^i(X,\mathbb {Q}) \to (\operatorname {H}^{2n-i-1}(U,\mathbb {Q}))^* \to \operatorname {H}^{i+1}(Z,\mathbb {Q}) \to \ldots .
From the functorial properties of the theory thus constructed, we obtain the useful corollaries [hodge-theory-iii-8.2.5 (?)] to [hodge-theory-iii-8.2.8 (?)], which we state in terms independent of any Hodge theory.
In [hodge-theory-iii-8.3 (?)], we endow the cohomology of any simplicial scheme X_\bullet with a mixed Hodge structure.
This generality is not illusory.
We can interpret relative cohomology spaces as being the cohomology of suitable simplicial schemes.
Let B_G be the classifying space of the underlying Lie group of an algebraic group G.
We can interpret the cohomology of B_G as being the cohomology of a suitable simplicial scheme.
In [hodge-theory-iii-9.1 (?)], after having calculated the mixed Hodge structure on the cohomology of B_G (for G linear), we thus obtain that of G.
As a corollary, we find that, if a linear group G acts on a non-singular complete variety X, then the map
\operatorname {H}^\bullet (X,\mathbb {Q}) \to \operatorname {H}^\bullet (X,\mathbb {Q})\otimes \operatorname {H}^\bullet (G,\mathbb {Q})
factors through \operatorname {H}^\bullet (X,\mathbb {Q})\otimes \operatorname {H}^0(G,\mathbb {Q})\subset \operatorname {H}^\bullet (X,\mathbb {Q})\otimes \operatorname {H}^\bullet (G,\mathbb {Q}).
In [hodge-theory-iii-10 (?)], we interpret in terms of abelian schemes the mixed Hodge structures of pure degree \{(-1,-1),(-1,0),(0,-1),(0,0)\} (see [hodge-theory-iii-0.5 (?)]), and we consider in detail the \operatorname {H}^1 of curves.
The theory developed up until here is an absolute theory (we do not study the functors \mathrm {R} f_*), and deals only with constant coefficients.
I conjecture that, if \mathscr {H} is a variation of polarisable Hodge structures (in the sense of Griffiths [G1970a]) on a scheme X, then the cohomology of X with coefficients in the local system \mathscr {H} is endowed with a natural mixed Hodge structure.
I can only prove this when X is complete.
2767hodge-theory-iii-terminology-and-notationhodge-theory-iii-terminology-and-notation.xmlTerminology and notationhodge-theory-iii1912hodge-theory-iii-0.1hodge-theory-iii-0.1.xmlIII.0.1hodge-theory-iii-terminology-and-notation
Let u\colon X\to Y be a continuous map between topological spaces.
We say that u is proper if it is proper in the sense of Bourbaki (i.e. universally closed) and furthermore separated (i.e. the diagonal of X\times _Y X is closed).
1913hodge-theory-iii-0.2hodge-theory-iii-0.2.xmlIII.0.2hodge-theory-iii-terminology-and-notation
Following Gabriel and Zisman, we say simplicial where we would have previously said semi-simplicial.
1914hodge-theory-iii-0.3hodge-theory-iii-0.3.xmlIII.0.3hodge-theory-iii-terminology-and-notation
We denote by A a Noetherian subring of \mathbb {R} such that A\otimes \mathbb {Q} is a field.
The useful cases are A=\mathbb {Z}, \mathbb {Q}, or \mathbb {R}.
1915hodge-theory-iii-0.4hodge-theory-iii-0.4.xmlIII.0.4hodge-theory-iii-terminology-and-notation
A mixed Hodge A-structure consists of an A-module H_A of finite type, a finite increasing filtration W on the A\otimes \mathbb {Q}-vector space H_{A\otimes \mathbb {Q}}=H_A\otimes \mathbb {Q}, and a finite decreasing filtration F on the \mathbb {C}-vector space H_\mathbb {C}=H_A\otimes _A\mathbb {C}.
We demand that the (\operatorname {Gr}_n^W(H_{A\otimes \mathbb {Q}}),\operatorname {Gr}_n^W(F)) be Hodge A\otimes \mathbb {Q}-structures.
For A=\mathbb {Z} (resp. A=\mathbb {Q}), we recover (resp. );
the results of carry over as they are.
1916hodge-theory-iii-0.5hodge-theory-iii-0.5.xmlIII.0.5hodge-theory-iii-terminology-and-notation
Let \mathscr {E} be a subset of \mathbb {Z}\times \mathbb {Z}.
We say that a mixed Hodge structure H is of degree \mathscr {E} if the Hodge numbers h^{pq} are zero for (p,q)\not \in \mathscr {E}.
1917hodge-theory-iii-0.6hodge-theory-iii-0.6.xmlIII.0.6hodge-theory-iii-terminology-and-notation
From now on, we denote by \Omega _X^\bullet (\log D) what we previously denoted by \Omega _X^\bullet \langle D\rangle ().
1918hodge-theory-iii-0.7hodge-theory-iii-0.7.xmlIII.0.7hodge-theory-iii-terminology-and-notation
Unless explicitly stated otherwise, scheme means "scheme of finite type over \mathbb {C}", and a sheaf on a scheme X is a sheaf on the underlying topological space of X_\mathrm {an}.
2771hodge-theory-iii-5hodge-theory-iii-5.xmlCohomological descent5hodge-theory-iii2768hodge-theory-iii-5.1hodge-theory-iii-5.1.xmlSimplicial topological spaces5.1hodge-theory-iii-5
This section starts with some reminders for which we can refer to the homotopical seminar of Strasbourg, 1963/64.
1937hodge-theory-iii-5.1.1hodge-theory-iii-5.1.1.xml5.1.1hodge-theory-iii-5.1
We will us the following notation, where n,k\geqslant -1 are integers.
\mathscr {A}^\circ
= the opposite category of a category \mathscr {A}
\underline {\operatorname {Hom}}(\mathscr {A},\mathscr {B})
= the category of functors from \mathscr {A} to \mathscr {B}.
\Delta _n
= the finite totally ordered set [0,n].
\delta _i\colon \Delta _n\to \Delta _{n+1}
= the increasing injection such that i\not \in \delta _i(\Delta _n) (for 0\leqslant i\leqslant n+1).
s_i\colon \Delta _{n+1}\to \Delta _n
= the increasing surjection such that s_i(i)=s_i(i+1) (for 0\leqslant i\leqslant n).
\varepsilon \colon \Delta _{-1}\to \Delta _n
= the unique map from \Delta _{-1} to \Delta _n.
(\Delta ^+)
= the category whose objects are the \Delta _n (for n\geqslant -1) and whose morphisms are the increasing maps between the \Delta _n.
(\Delta )
= the full subcategory of (\Delta ^+) whose objects are the \Delta _n (for n\geqslant 0).
(\Delta ^+)_k
= the full subcategory of (\Delta ^+) whose objects are the \Delta _n (for k\geqslant n).
(\Delta )_k
= the full subcategory of (\Delta ^+) whose objects are the \Delta _n (for k\geqslant n\geqslant 0).
For any category \mathscr {C}, we define a simplicial object (resp. k-truncated simplicial object) of \mathscr {C} to be an object of \underline {\operatorname {Hom}}((\Delta )^0,\mathscr {C}) (resp. of \underline {\operatorname {Hom}}((\Delta )^0_k,\mathscr {C})).
Similarly, a cosimplicial object (resp. k-truncated cosimplicial object) is an object of \underline {\operatorname {Hom}}((\Delta ),\mathscr {C}) (resp. of \underline {\operatorname {Hom}}((\Delta )_k,\mathscr {C})).
For k\geqslant -1, the k-skeleton functor is the restriction functor
\operatorname {sq}_k\colon \underline {\operatorname {Hom}}((\Delta )^0,\mathscr {C}) \to \underline {\operatorname {Hom}}((\Delta )_k^0,\mathscr {C})
and the k-coskeleton functor
\operatorname {cosq}_k\colon \underline {\operatorname {Hom}}((\Delta )_k^0,\mathscr {C}) \to \underline {\operatorname {Hom}}((\Delta )^0,\mathscr {C}).
is the right adjoint to \operatorname {sq}_k.
Let Y be a simplicial object of \mathscr {C}.
We also call the functor
\operatorname {sq}_k\colon \underline {\operatorname {Hom}}((\Delta )^0,\mathscr {C})/Y \to \underline {\operatorname {Hom}}((\Delta )_k^0,\mathscr {C})/\operatorname {sq}_k(Y)
the skeleton, and its right adjoint
\operatorname {cosq}_k^Y\colon \underline {\operatorname {Hom}}((\Delta )_k^0,\mathscr {C})/\operatorname {sq}_k(Y) \to \underline {\operatorname {Hom}}((\Delta )^0,\mathscr {C})/Y
is called the coskeleton with respect to Y.
The coskeleton functors exist if finite projective limits exist in \mathscr {C};
the relative coskeleton functors exist if fibred products do;
we have
\operatorname {cosq}_k^Y(X) = \operatorname {cosq}_k(X) \times _{\operatorname {cosq}_k\operatorname {sq}_k(Y)}Y. 1938hodge-theory-iii-5.1.2hodge-theory-iii-5.1.2.xml5.1.2hodge-theory-iii-5.1
If X_\bullet \colon (\Delta )^0\to \mathscr {C} is a simplicial object of \mathscr {C}, then we set
X_n = X_\bullet (\Delta _n)
\delta _i = X_\bullet (\delta _i\colon \Delta _n\to \Delta _{n+1})\colon X_{n+1}\to X_n
s_i = X_\bullet (s_i\colon \Delta _{n+1}\to \Delta _n)\colon X_n\to X_{n+1}.
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
X_\bullet = \Big (\ldots \quad
X_2
\ar [r,"\delta _1" description]
\ar [r,shift left=6,"\delta _0" description]
\ar [r,shift right=6,"\delta _2" description]
& X_1
\ar [r,shift left=3,"\delta _0" description]
\ar [r,shift right=3,"\delta _1" description]
\ar [l,shift right=3,"s_0" description]
\ar [l,shift left=3,"s_1" description]
& X_0\Big )
\ar [l,"s_0" description]
\end {tikzcd}1939hodge-theory-iii-5.1.3hodge-theory-iii-5.1.3.xml5.1.3hodge-theory-iii-5.1
Let S\in \operatorname {Ob}\mathscr {C}.
The constant simplicial object S_\bullet is the simplicial object for which S_n=S and \delta _i=s_i=\mathrm {Id}_S.
A simplicial (resp. k-truncated simplicial) object of \mathscr {C} augmented over S is a morphism a\colon X_\bullet \to S_\bullet (resp. a\colon X_\bullet \to \operatorname {sq}_k(S_\bullet )).
We identify such a simplicial object (resp. k-truncated simplicial object) augmented over S with the object X_\bullet ^+ of \underline {\operatorname {Hom}}((\Delta ^+)^0,\mathscr {C}) (resp. of \underline {\operatorname {Hom}}((\Delta ^+)_k^0,\mathscr {C})) such that X_\bullet ^+(\Delta _{-1})=S.
We have that a_n=X_\bullet ^+(\varepsilon \colon \Delta _{-1}\to \Delta _n).
These objects will be denoted by notation of the form a\colon X_\bullet \to S.
The relative coskeletons will be mostly used in this setting, and denoted \operatorname {cosq}_k^S or simply \operatorname {cosq}_k.
1940hodge-theory-iii-5.1.4hodge-theory-iii-5.1.4.xml5.1.4hodge-theory-iii-5.1
The coskeleton of the augmented 0-truncated simplicial object a_0\colon X\to S is the simplicial object augmented over S whose components are the cartesian powers of X in \mathscr {C}/S, i.e.
\operatorname {cosq}_0(X\to S) = \big ( ((X/S)^{\Delta _n})_{n\geqslant 0} \to S \big ). 1942hodge-theory-iii-5.1.5hodge-theory-iii-5.1.5.xml5.1.5hodge-theory-iii-5.1
Let u\colon X\to Y be a continuous map between topological spaces, and F a sheaf on X and G a sheaf on Y.
The set \operatorname {Hom}_u(G,F) of u-morphisms from G to F is the set \operatorname {Hom}(u^*G,F)\cong \operatorname {Hom}(G,u_*F).
The data of a u-morphism f from G to F consists of the data of, for each pair of opens (U\subset X,V\subset Y) such that u(U)\subset V, a map f_{UV}\colon G(V)\to F(U);
these maps must satisfy the condition
1941Conditionhodge-theory-iii-5.1.5.starhodge-theory-iii-5.1.5.star.xml*hodge-theory-iii-5.1.5
For U'\subset U\subset X and V'\subset V\subset Y such that u(U)\subset V and u(U')\subset V', the diagram
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
G(V)
\ar [r]
\ar [d]
& G(V')
\ar [d]
\\F(U)
\ar [r]
& F(U')
\end {tikzcd}
commutes.
1946hodge-theory-iii-5.1.6hodge-theory-iii-5.1.6.xml5.1.6hodge-theory-iii-5.1
A simplicial topological space is a simplicial object of the category whose objects are topological spaces and whose morphisms are continuous maps.
A sheaf F^\bullet on a simplicial topological space X_\bullet consists of
a family of sheaves F^n on the X_n;
for all f\colon \Delta _n\to \Delta _m, an X_\bullet (f)-morphism F^\bullet (f) from F^n to F^m.
We further require that F^\bullet (f\circ g)=F^\bullet (f)\circ F^\bullet (g).
A morphism u from F^\bullet to G^\bullet is a family of morphisms u^n\colon F^n\to G^n such that, for all f\in \operatorname {Hom}_{(\Delta )}(\Delta _n,\Delta _m), we have that u^m\circ F^\bullet (f)=G^\bullet (f)\circ u^n.
1950hodge-theory-iii-5.1.7hodge-theory-iii-5.1.7.xml5.1.7hodge-theory-iii-5.1
For an open U of X_n, let F^\bullet (U)=F^n(U).
For f\colon \Delta _n\to \Delta _m, U\subset X_n, and V\subset X_m such that X_\bullet (f)(V)\subset (U), let F^\bullet (f,V,U)\colon F^\bullet (U)\to F^\bullet (V) be the map induced by F^\bullet (f).
We immediately see that the system of sets F^\bullet (U) (indexed by n\geqslant 0 and U\subset X_n) and of maps F^\bullet (f,V,U) uniquely determines F^\bullet (cf. ).
For such a system to come from a sheaf on X_\bullet , it is necessary and sufficient that:
F^\bullet (fg,U,W)=F^\bullet (f,U,V)F^\bullet (g,V,W) whenever this makes sense;
for any n, the F^\bullet (U) for U\subset X_n form a sheaf on X_n.
1951hodge-theory-iii-5.1.8hodge-theory-iii-5.1.8.xml5.1.8hodge-theory-iii-5.1
This shows that sheaves on X_\bullet can be interpreted as sheaves on a suitable site, and, in particular, form a topos (X_\bullet )^\sim .
We freely use, for sheaves on X_\bullet , the modern terminology for sheaves on a site.
With this in mind, a sheaf F^\bullet of abelian groups (resp. of rings, ...) is a system of sheaves F^n of abelian groups (resp. of rings, ...) on the X_n along with the morphisms F^\bullet (f).
1960Exampleshodge-theory-iii-5.1.9hodge-theory-iii-5.1.9.xml5.1.9hodge-theory-iii-5.1
Let X_\bullet be a simplicial analytic space.
The structure sheaves \mathcal {O}_{X_n} form a sheaf of rings on X_\bullet .
Let X_\bullet be a simplicial analytic space augmented over S.
The sheaves \Omega _{X_n/S}^1 form a sheaf of \mathcal {O}-modules on X_\bullet .
Its i-th exterior power is the sheaf of \mathcal {O}-modules (\Omega _{X_n/S}^i)_{n\geqslant 0} denoted \Omega _{X_\bullet {/}S}^i.
The de Rham complexes (\Omega _{X_n/S}^\bullet )_{n\geqslant 0} form a complex of sheaves on X_\bullet .
Let F^\bullet be a sheaf of abelian groups on X_\bullet .
The canonical flasque Godement resolutions \mathscr {C}^\bullet (F^n) form a complex of sheaves on X_\bullet that is a resolution of F^\bullet .
Let S be a topological space.
Sheaves F^\bullet on the constant simplicial space S_\bullet can be identified with cosimplicial sheaves on S^\bullet .
In particular, an abelian sheaf F^\bullet on S_\bullet defines a chain complex (F^n,d=\sum _i(-1)^i\delta _i).
A complex K of abelian sheaves on S_\bullet defines a double complex (K^{n,m}) that we again denote by K (here m is the cosimplicial degree) and whose associated simple complex \mathrm {s}{K} is
1957Equationhodge-theory-iii-5.1.9.1hodge-theory-iii-5.1.9.1.xml5.1.9.1hodge-theory-iii-5.1.9 (\mathrm {s}{K})^n = \bigoplus _{p+q=n} K^{pq} \tag{5.1.9.1}
with differential given by
1958Equationhodge-theory-iii-5.1.9.2hodge-theory-iii-5.1.9.2.xml5.1.9.2hodge-theory-iii-5.1.9 \mathrm {d}(x^{pq}) = \mathrm {d}_K(x^{pq}) + (-1)^p\sum _i(-1)^i\delta _i x^{pq}. \tag{5.1.9.2}
We denote by L the second filtration of \mathrm {s}{K}, i.e.
1959Equationhodge-theory-iii-5.1.9.3hodge-theory-iii-5.1.9.3.xml5.1.9.3hodge-theory-iii-5.1.9 L^r(\mathrm {s}{K}) = \bigoplus _{q\geqslant r} K^{pq}. \tag{5.1.9.3}
1961hodge-theory-iii-5.1.10hodge-theory-iii-5.1.10.xml5.1.10hodge-theory-iii-5.1
Let u\colon X_\bullet \to Y_\bullet be a morphism of simplicial topological spaces, with components u_n\colon X_n\to Y_n.
If G (resp. F) is a sheaf on Y_\bullet (resp. on X_\bullet ), then (u_n^*G^n)_{n\geqslant 0} (resp. ({u_n}_*F^n)_{n\geqslant 0}) is a sheaf on X_\bullet (resp. on Y_\bullet );
we denote it by u^*G (resp. by u_*F).
The functors u^* and u_* are adjoint to one another;
they are the inverse image and direct image morphisms of topos morphism u\colon (X_\bullet )^\sim \to (Y_\bullet )^\sim .
1963hodge-theory-iii-5.1.11hodge-theory-iii-5.1.11.xml5.1.11hodge-theory-iii-5.1
Let a\colon X_\bullet \to S be an augmented simplicial topological space.
If F is a sheaf on S, then a^*F=(a_n^*F)_{n\geqslant 0} "is" a sheaf on X_\bullet .
The functor a^* has a right adjoint
1962Equationhodge-theory-iii-5.1.11.1hodge-theory-iii-5.1.11.1.xml5.1.11.1hodge-theory-iii-5.1.11 a_*\colon F^\bullet \mapsto \operatorname {Ker}\big ( {a_0}_* F^0 \,\overset {\delta _0^*}{\underset {\delta _1^*}{\rightrightarrows }}\, {a_1}_* F^1 \big ). \tag{5.1.11.1}
The functors a^* and a_* are the inverse image and direct image morphisms of topos morphism a\colon (X_\bullet )^\sim \to (S)^\sim .
1966hodge-theory-iii-5.1.12hodge-theory-iii-5.1.12.xml5.1.12hodge-theory-iii-5.1
Let S_\bullet be the constant simplicial space associated to S, and a_\bullet \colon X_\bullet \to S_\bullet the morphism defined by a.
For an abelian sheaf F^\bullet on X_\bullet , we can identify {a_\bullet }_*F^\bullet with a cosimplicial sheaf on S ((IV) of ).
For a complex K of abelian sheaves on X_\bullet , if K^{pq} is the degree p component of the restriction of K to X_q, then the components of the simple complex associated to the double complex defined by {a_\bullet }_*K are
1964Equationhodge-theory-iii-5.1.12.1hodge-theory-iii-5.1.12.1.xml5.1.12.1hodge-theory-iii-5.1.12 (\mathrm {s}{a_\bullet }_*K)^n = \bigoplus _{p+q=n} {a_q}_* K^{pq}. \tag{5.1.12.1}
The spectral sequence defined by the filtration L () of \mathrm {s}{a_\bullet }_*K can be written as
1965Equationhodge-theory-iii-5.1.12.2hodge-theory-iii-5.1.12.2.xml5.1.12.2hodge-theory-iii-5.1.12 E_1^{pq} = \operatorname {H}^q({a_p}_*(K|X^p)) \Rightarrow \operatorname {H}^{p+q}(\mathrm {s}{a_\bullet }_*K). \tag{5.1.12.2} 1968hodge-theory-iii-5.1.13hodge-theory-iii-5.1.13.xml5.1.13hodge-theory-iii-5.1
Let S=P^t (the topological space consisting of a single point).
We see that, for a sheaf F^\bullet on X_\bullet , the \Gamma (X_n,F^n) are the components of a cosimplicial set \Gamma ^\bullet (X_\bullet ,F^\bullet ).
The functor \Gamma (of global sections) is the functor
1967Equationhodge-theory-iii-5.1.13.1hodge-theory-iii-5.1.13.1.xml5.1.13.1hodge-theory-iii-5.1.13 \Gamma \colon F^\bullet \mapsto \operatorname {Ker}\big ( \Gamma (X_0,F^0) \rightrightarrows \Gamma (X_1,F^1) \big ). \tag{5.1.13.1}
If K is a complex of abelian sheaves, we denote by \Gamma ^\bullet (X_\bullet ,K) the chain complex whose components are the cosimplicial abelian groups \Gamma ^\bullet (X_\bullet ,K^n), and by \mathrm {s} \Gamma ^\bullet (X_\bullet ,K) the associated simple complex.
2769hodge-theory-iii-5.2hodge-theory-iii-5.2.xmlCohomology of simplicial topological spaces5.2hodge-theory-iii-52040hodge-theory-iii-5.2.1hodge-theory-iii-5.2.1.xml5.2.1hodge-theory-iii-5.2
To each simplicial topological space X_\bullet is associated a usual topological space |X_\bullet |, called its geometric realisation (see below).
In the particular case where the X_n are discrete, we recover the usual notion of geometric realisation of a simplicial set.
In [S1968], G. Segal defined the cohomology of X_\bullet with values in an abelian group A as \operatorname {H}^\bullet (|X_\bullet |,A).
Under suitable hypotheses, the filtration of |X_\bullet | by successive skeletons gives a spectral sequence
352Equationhodge-theory-iii-5.2.1.1hodge-theory-iii-5.2.1.1.xml5.2.1.1hodge-theory-iii-5.2.1 E_1^{pq} = \operatorname {H}^q(X_p,A) \Rightarrow \operatorname {H}^{p+q}(X_\bullet ,A). \tag{5.2.1.1}
One of the interests in using this definition is that it also applies, for example, to the definition of the groups K(X_\bullet ).
We will adopt another definition, which is better adapted to sheaf-theoretic techniques.
353#179unstable-179.xmlGeometric realisationhodge-theory-iii-5.2.1
For an integer n\geqslant 0, we denote by |\Delta _n| the simple in \mathbb {R}^{\Delta _n} whose vertices are the set of basis vectors.
We identify \Delta _n with the set of vertices of |\Delta _n|.
Every function f\colon \Delta _n\to \Delta _m extends by linearity to |f|\colon |\Delta _n|\to |\Delta _m|.
Let X_\bullet be a simplicial topological space.
Let
Y=\coprod _{n\geqslant 0}(X_n\times |\Delta _n|).
Let R be the finest equivalence relation on Y for which, for any f\colon \Delta _n\to \Delta _m in (\Delta ), any x_m\in X_m, and any a\in |\Delta _n|, we have
(x_m,|f|(a)) \equiv (f(x_m),a) \operatorname {mod} R.
The geometric realisation of X_\bullet is by definition
|X_\bullet | = Y/R. 2041Definitionhodge-theory-iii-5.2.2hodge-theory-iii-5.2.2.xml5.2.2hodge-theory-iii-5.2
Let X_\bullet be a simplicial topological space.
The functors \operatorname {H}^i(X_\bullet ,F^\bullet ) of cohomology with coefficients in the abelian sheaf F^\bullet on X_\bullet are the derived functors of the functor \Gamma ().
This definition is equivalent to the following, sometimes more convenient.
2042hodge-theory-iii-5.2.3hodge-theory-iii-5.2.3.xml5.2.3hodge-theory-iii-5.2
Let F^\bullet be an abelian sheaf on X_\bullet .
We can show (by example, using (III) of ) that F^\bullet always admits resolutions K on the right such that
\operatorname {H}^r(X_q,K^{p,q}) =0 \quad \text {for } p,q\geqslant 0 \text { and } r>0.
If K is such a resolution, then we canonically have that
1984Equationhodge-theory-iii-5.2.3.1hodge-theory-iii-5.2.3.1.xml5.2.3.1hodge-theory-iii-5.2.3 \operatorname {H}^n(X_\bullet ,F^\bullet ) \simeq \operatorname {H}^n(\mathrm {s}\Gamma ^\bullet (X_\bullet ,K)). \tag{5.2.3.1}
It is easy to show directly that the right-hand side is independent (up to unique isomorphism) of the choice of K;
for what follows, we are free to define \operatorname {H}^\bullet (X_\bullet ,F^\bullet ) by .
We can further show that the spectral sequence of the complex \mathrm {s}\Gamma ^\bullet (X_\bullet ,K), filtered by L ( and for S=P^t)
1985Equationhodge-theory-iii-5.2.3.2hodge-theory-iii-5.2.3.2.xml5.2.3.2hodge-theory-iii-5.2.3 E_1^{pq} = \operatorname {H}^q(X_p,F^p) \Rightarrow \operatorname {H}^{p+q}(X_\bullet ,F^\bullet ) \tag{5.2.3.2}
is independent (up to unique isomorphism) of the choice of K (cf. ).
2045hodge-theory-iii-5.2.4hodge-theory-iii-5.2.4.xml5.2.4hodge-theory-iii-5.2
We will need to make precise the above construction when passing to derived categories, and to give a relative variant of it.
Let a\colon X_\bullet \to S, let S_\bullet be the constant simplicial space defined by S with augmentation map \varepsilon \colon S_\bullet \to S, and let a_\bullet \colon X_\bullet \to S_\bullet be the map induced by a.
Then
2043Equationhodge-theory-iii-5.2.4.1hodge-theory-iii-5.2.4.1.xml5.2.4.1hodge-theory-iii-5.2.4 a_* = \varepsilon _* {a_\bullet }_*\colon (X_\bullet )^\sim \to (S)^\sim . \tag{5.2.4.1}
The derived version of this equation is
2044Equationhodge-theory-iii-5.2.4.2hodge-theory-iii-5.2.4.2.xml5.2.4.2hodge-theory-iii-5.2.4 \mathrm {R} a_* = \mathrm {R}\varepsilon _* \mathrm {R}{a_\bullet }_*\colon \mathrm {D}^+(X_\bullet ) \to \mathrm {D}^+(S) \to (S)^\sim . \tag{5.2.4.2}
where \mathrm {D}^+(X_\bullet ) is the bounded-below derived category of the category of abelian sheaves on X_\bullet .
We will calculate \mathrm {R}{a_\bullet }_* and \mathrm {R}\varepsilon _*.
2046hodge-theory-iii-5.2.5hodge-theory-iii-5.2.5.xml5.2.5hodge-theory-iii-5.2
Let u\colon X_\bullet \to Y_\bullet be a morphism of simplicial topological spaces.
The functor \mathrm {R} u_*\colon \mathrm {D}^+(X_\bullet )\to \mathrm {D}^+(Y_\bullet ) can be calculated "component by component": if K is a complex of abelian sheaves on X_\bullet , then to calculate \mathrm {R} u_*K we take a resolution K\xrightarrow {\sim } K' such that the components F of K' satisfy \mathrm {R}^i {u_p}_*(F^p)=0 for i>0, p\geqslant 0 (cf. (III) of );
then \mathrm {R} u_*K\approx u_*K'.
2048hodge-theory-iii-5.2.6hodge-theory-iii-5.2.6.xml5.2.6hodge-theory-iii-5.2
The functor \mathrm {s}\colon \operatorname {C}^+(S_\bullet )\to \operatorname {C}^+(S) sends acyclic complexes to acyclic complexes.
It thus trivially derives to
\mathrm {s}\colon \mathrm {D}^+(S_\bullet ) \to \mathrm {D}^+(S).
If F is an injective sheaf on S_\bullet , then the chain complex \mathrm {s} F is a resolution of \varepsilon _*F.
We thus obtain an isomorphism \mathrm {R}\varepsilon _*\xrightarrow {\sim }\mathrm {s}, and becomes
2047Equationhodge-theory-iii-5.2.6.1hodge-theory-iii-5.2.6.1.xml5.2.6.1hodge-theory-iii-5.2.6 \mathrm {R} a_* = \mathrm {s}\mathrm {R}{a_\bullet }_* \tag{5.2.6.1} 2049hodge-theory-iii-5.2.7hodge-theory-iii-5.2.7.xml5.2.7hodge-theory-iii-5.2
Combined with and specialised to the case where S=P^t, this equation proves .
In concrete terms, this implies that, to calculate \mathrm {R} a_*K, we can proceed in two steps:
We take a resolution K\xrightarrow {\sim } K' such that the components F of K' satisfy \mathrm {R}^i {a_p}_*(F^p)=0 for i>0.
The complex {a_\bullet }_*K'\in \mathrm {D}^+(S_\bullet ) (the derived category of the category of cosimplicial abelian sheaves on S) can be identified with \mathrm {R}{a_\bullet }_*K.
\mathrm {R} a_*K is the simple complex \mathrm {s}{a_\bullet }_*K' associated to the double complex {a_\bullet }_*K'.
The spectral sequence generalises to a spectral sequence
1990Equationhodge-theory-iii-5.2.7.1hodge-theory-iii-5.2.7.1.xml5.2.7.1hodge-theory-iii-5.2.7 E_1^{pq} = \mathrm {R}^q{a_p}_*(K|X_p) \Rightarrow \mathrm {R}^{p+q}a_*K \tag{5.2.7.1}
induced by .
2770hodge-theory-iii-5.3hodge-theory-iii-5.3.xmlCohomological descent5.3hodge-theory-iii-52105hodge-theory-iii-5.3.1hodge-theory-iii-5.3.1.xml5.3.1hodge-theory-iii-5.3
Let a\colon X_\bullet \to S be an augmented simplicial topological space.
For every sheaf F on S, we have a morphism
\varphi \colon F \to a_*a^*F
from the adjunction a^*\dashv a_*.
This morphism derives to a morphism of functors from \mathrm {D}^+(S) to \mathrm {D}^+(S), namely
2104Equationhodge-theory-iii-5.3.1.1hodge-theory-iii-5.3.1.1.xml5.3.1.1hodge-theory-iii-5.3.1 \varphi \colon \mathrm {Id} \to \mathrm {R} a_*a^*. \tag{5.3.1.1} 2106Definitionhodge-theory-iii-5.3.2hodge-theory-iii-5.3.2.xml5.3.2hodge-theory-iii-5.3
We say that a is of cohomological descent if, for every abelian sheaf F on S, we have
F\xrightarrow {\sim }\operatorname {Ker}({a_0}_*{a_0}^*F\to {a_1}_*{a_1}^*F)
and
\mathrm {R}^i a_*a^*F = 0 \quad \text {for }i>0.
This is equivalent to asking that be an isomorphism.
2107hodge-theory-iii-5.3.3hodge-theory-iii-5.3.3.xml5.3.3hodge-theory-iii-5.3
If a is of cohomological descent, then, for K\in \operatorname {Ob}\mathrm {D}^+(S), the canonical map
2071Equationhodge-theory-iii-5.3.3.1hodge-theory-iii-5.3.3.1.xml5.3.3.1hodge-theory-iii-5.3.3 \mathrm {R}\Gamma (S,K) \to \mathrm {R}\Gamma (S,\mathrm {R} a_*a^*K) \simeq \mathrm {R}\Gamma (X_\bullet ,a^*K) \tag{5.3.3.1}
is an isomorphism.
In particular, for F an abelian sheaf on S, we have a spectral sequence ()
2072Equationhodge-theory-iii-5.3.3.2hodge-theory-iii-5.3.3.2.xml5.3.3.2hodge-theory-iii-5.3.3 E_1^{pq} = \operatorname {H}^q(X_p,a_p^*F) \Rightarrow \operatorname {H}^{p+q}(S,F). \tag{5.3.3.2}
For a complex, we again have, in hypercohomology, a spectral sequence
2073Equationhodge-theory-iii-5.3.3.3hodge-theory-iii-5.3.3.3.xml5.3.3.3hodge-theory-iii-5.3.3 E_1^{pq} = \operatorname {\mathbb {H}}^q(X_p,a_p^*K) \Rightarrow \operatorname {\mathbb {H}}^{p+q}(S,K). \tag{5.3.3.3}
In both cases, the E_1^{\bullet q} (for fixed q) form a simplicial group, and
d_1 = \sum _i (-1)^i\delta _i \colon E_1^{p,q} \to E_1^{p+1,q}. 2108Definitionhodge-theory-iii-5.3.4hodge-theory-iii-5.3.4.xml5.3.4hodge-theory-iii-5.3
A continuous map a\colon X\to S is of cohomological descent if the augmentation morphism of \operatorname {cosq}(X\to S), namely
\big ((X/S)^{\Delta _n}\big )_{n\geqslant 0} \to S,
is of cohomological descent.
We say that a is of universal cohomological descent if, for every u\colon S'\to S, the continuous map a'\colon X\times _S S'\to S' is of cohomological descent.
2109hodge-theory-iii-5.3.5hodge-theory-iii-5.3.5.xml5.3.5hodge-theory-iii-5.3
The fundamental results, proven in [SD], are the following.
The continuous maps of universal cohomological descent form a Grothendieck topology on the category of topological spaces, which we call the universal cohomological descent topology.
A proper () surjective map is of universal cohomological descent.
A map a\colon X\to S that admits sections locally on S is of universal cohomological descent.
Let a\colon X_\bullet \to S be a k-truncated augmented simplicial space (with -1\leqslant k\leqslant \infty ).
For k\geqslant n\geqslant -1, let \varphi _n\colon \operatorname {cosq} X_\bullet \to \operatorname {cosq}\operatorname {sq}_n X_\bullet be the evident map.
We say that X_\bullet is a k-truncated hypercover of S, for the universal cohomological descent topology, if the maps
1909Equationhodge-theory-iii-5.3.5.1hodge-theory-iii-5.3.5.1.xml5.3.5.1hodge-theory-iii-5.3.5 (\varphi _n)_{n+1} \colon X_{n+1} \to (\operatorname {cosq}\operatorname {sq}_n X_\bullet )_{n+1} \tag{5.3.5.1}
(for -1\leqslant n\leqslant k-1) are of universal cohomological descent.
If X_\bullet is such a hypercover, then the simplicial space \operatorname {cosq}(X_\bullet ) augmented over S is of cohomological descent.
Let a be a morphism of simplicial topological spaces augmented over S.
\begin {CD} X_\bullet @>a>> Y_\bullet \\@VxVV @VVyV \\S @= S \end {CD}
We say that a is a hypercover for the universal cohomological descent topology if the evident maps X_n\to (\operatorname {cosq}_{n-1}^{Y_\bullet }\operatorname {sq}_{n-1}X_\bullet )_n are of universal cohomological descent.
If a is such a hypercover then, for every K\in \operatorname {Ob}\mathrm {D}^+(S),
a^*\colon \mathrm {R} y_*y^*K \xrightarrow {\sim } \mathrm {R} x_*x^*K.
2110hodge-theory-iii-5.3.6hodge-theory-iii-5.3.6.xml5.3.6hodge-theory-iii-5.3
For k=\infty , (IV) of implies that the a\colon X_\bullet \to S are of cohomological descent if the (\varphi _n)_{n+1} are of universal cohomological descent.
For n=-1,0, these maps are
(\varphi _n)_{n+1} = \begin {cases} X_0\xrightarrow {a}S &\text {if }n=-1 \\X_1\xrightarrow {(\delta _0,\delta _1)}X_0\times _S X_0 &\text {if }n=0. \end {cases}
For n=1, (\operatorname {cosq}\operatorname {sq}_1(X_\bullet ))_1 is the subspace of X_1\times _S X_1\times _S X_1 consisting of the triples (x,y,z) such that \delta _0x=\delta _0y, \delta _1x=\delta _0z, and \delta _1y=\delta _1z.
The map (\varphi _1)_2 is x\mapsto (\delta _0x,\delta _1x,\delta _2x).
For k=0, (IV) of is .
2111Examplehodge-theory-iii-5.3.7hodge-theory-iii-5.3.7.xml5.3.7hodge-theory-iii-5.3
Let \mathscr {U}=(U_i)_{i\in I} be an open cover, or a finite locally closed cover, of S.
Let X=\coprod _{i\in I}U_i.
Then a\colon X\to S is of cohomological descent.
The spectral sequence in for X_\bullet =\operatorname {cosq}(X\to S) is then exactly the Leray spectral sequence of the cover \mathscr {U}.
2112hodge-theory-iii-5.3.8hodge-theory-iii-5.3.8.xml5.3.8hodge-theory-iii-5.3
Let a\colon X_\bullet \to S be as in (IV) of .
We say that X_\bullet is a proper k-truncated hypercover of S if the arrows in are proper and surjective.
For k=\infty , we simply say "proper hypercover".
2772hodge-theory-iii-6hodge-theory-iii-6.xmlExamples of simplicial topological spaces6hodge-theory-iii2751hodge-theory-iii-6.1hodge-theory-iii-6.1.xmlClassifying spaces6.1hodge-theory-iii-62172hodge-theory-iii-6.1.1hodge-theory-iii-6.1.1.xml6.1.1hodge-theory-iii-6.1
Let u\colon X\to S be a continuous map.
For every sheaf F on S, the sheaf u^*F is endowed with a "descent data" with respect to u, i.e. we have an isomorphism between the two inverse images of u^*F on X\times _S X, and this isomorphism satisfies a cocycle condition.
If u admits a section locally on S, then this construction defines an equivalence between the category of sheaves on S and that of sheaves on X endowed with a descent data.
Take X to be a (left) principal homogeneous space for the group G on a space S.
Then the G-equivariant sheaves on X are exactly the sheaves endowed with a descent data: every equivariant sheaf on X is, in a unique way (as an equivariant sheaf), the inverse image of a sheaf on S=X/G.
2173hodge-theory-iii-6.1.2hodge-theory-iii-6.1.2.xml6.1.2hodge-theory-iii-6.1
If a topological group G acts on a space X, then G acts on G^{\Delta _n}\times X by
g\cdot (g_0,\ldots ,g_n,x) = (g_0g^{-1},\ldots ,g_ng^{-1},gx).
We denote by [X/G]_\bullet the simplicial space
2138Equationhodge-theory-iii-6.1.2.1hodge-theory-iii-6.1.2.1.xml6.1.2.1hodge-theory-iii-6.1.2 [X/G]_\bullet = \big ((G^{\Delta _n}\times X)/G\big )_{n\geqslant 0}. \tag{6.1.2.1}
If X is a principal homogeneous space for the group G on S=X/G, then the map
\begin {aligned} G^{\Delta _n}\times X &\to X^{\Delta _n} \\(g_0,\ldots ,g_n,x) &\mapsto (g_0x,\ldots ,g_nx) \end {aligned}
identifies [X/G]_n with the iterated fibre product (X/S)^{\Delta _n}:
\operatorname {cosq}(X\to S) = ([X/G]_\bullet \to S).
In particular, we have (by (III) of )
2141Equationhodge-theory-iii-6.1.2.2hodge-theory-iii-6.1.2.2.xml6.1.2.2hodge-theory-iii-6.1.2 \operatorname {H}^\bullet ([X/G]_\bullet ) \simeq \operatorname {H}^\bullet (X/G) \tag{6.1.2.2}
(for a principal homogeneous space).
For all n, G^{\Delta _n}\times X is a principal homogeneous space for the group G on [X/G]_n.
For every equivariant sheaf F on X, \operatorname {pr}_2^*F is an equivariant sheaf on G^{\Delta _n}\times X;
by , the latter is the inverse image of F^n on [X/G]_n.
Every equivariant sheaf F on X thus defines a sheaf on [X/G]_\bullet .
It is easy to show that we thus obtain an equivalence between the category of equivariant sheaves on X and the category of sheaves F^\bullet on [X/G]_\bullet that satisfy
2143Propertyhodge-theory-iii-6.1.starhodge-theory-iii-6.1.star.xml*hodge-theory-iii-6.1.2(*) For every f\colon \Delta _n\to \Delta _m, the structure morphism f^*F^n\to F^m is an isomorphism.
Construction (b) above is natural in (G,X,F).
We set
2145Equationhodge-theory-iii-6.1.2.3hodge-theory-iii-6.1.2.3.xml6.1.2.3hodge-theory-iii-6.1.2 \operatorname {H}^\bullet (X,G;F) = \operatorname {H}^\bullet ([X/G]_\bullet ,F^\bullet ) \tag{6.1.2.3}
(mixed cohomology of X,G with coefficients in F).
Under the hypotheses of (a), if F is the inverse image of F^{-1} on S=X/G, then
2146Equationhodge-theory-iii-6.1.2.4hodge-theory-iii-6.1.2.4.xml6.1.2.4hodge-theory-iii-6.1.2 \operatorname {H}^\bullet (X,G;F) = \operatorname {H}^\bullet (X/G,F^{-1}) \tag{6.1.2.4}
(for a principal homogeneous space).
This generalises (which is the case F=\underline {\mathbb {Z}}).
2175hodge-theory-iii-6.1.3hodge-theory-iii-6.1.3.xml6.1.3hodge-theory-iii-6.1
Let P^t be the topological space consisting of a single point.
We define the simplicial classifying space of G, denoted B_{\bullet G}, to be the simplicial space
B_{\bullet G} = [P^t/G]_\bullet .
Let P be a principal homogeneous space for the group G on S.
The evident morphism
\operatorname {cosq}(P\to S) = [P/G]_\bullet \to [P^t/G]_\bullet = B_{\bullet G}
defines a composite morphism
2174Equationhodge-theory-iii-6.1.3.1hodge-theory-iii-6.1.3.1.xml6.1.3.1hodge-theory-iii-6.1.3 [P] \colon \operatorname {H}^\bullet (B_{\bullet G}) \to \operatorname {H}^\bullet ([P/G]_\bullet ) \xleftarrow [(6.1.2.2)]{\sim } \operatorname {H}^\bullet (S). \tag{6.1.3.1}
We will see below that, in good cases, \operatorname {H}^\bullet (B_{\bullet G})=\operatorname {H}^\bullet (B_G), and that the image of [P] consists of the characteristic classes of P.
2178hodge-theory-iii-6.1.4hodge-theory-iii-6.1.4.xml6.1.4hodge-theory-iii-6.1
Let G be a Lie group, B_G a classifying space for G, and a\colon U_G\to B_G the universal principal homogeneous G-space.
Let X be a G-space;
then X\times U_G is a principal homogeneous G-space over X\times U_G/G such that, for every equivariant sheaf F on X, \operatorname {pr}_1^*F is the inverse image of a sheaf F^G on X\times U_G/G.
Since U_G is contractible, and by , we have
2176Equationhodge-theory-iii-6.1.4.1hodge-theory-iii-6.1.4.1.xml6.1.4.1hodge-theory-iii-6.1.4 \operatorname {H}^\bullet (X,G;F) \xrightarrow {\sim } \operatorname {H}^\bullet (X\times U_G,G;\operatorname {pr}_1^*F) \xleftarrow {\sim } \operatorname {H}^\bullet (X\times U_G/G,F^G). \tag{6.1.4.1}
In particular, for X=P^t,
2177Equationhodge-theory-iii-6.1.4.2hodge-theory-iii-6.1.4.2.xml6.1.4.2hodge-theory-iii-6.1.4 \operatorname {H}^\bullet (B_{\bullet G}) = \operatorname {H}^\bullet (B_G). \tag{6.1.4.2}
We can see that the isomorphism in is a particular case of where S=B_G and P=U_G.
2179hodge-theory-iii-6.1.5hodge-theory-iii-6.1.5.xml6.1.5hodge-theory-iii-6.1
The spectral sequence in for B_{\bullet G}
E_1^{pq} = \operatorname {H}^q(G^{\Delta _p}/G) \Rightarrow \operatorname {H}^{p+q}(B_{\bullet G}) =\operatorname {H}^{p+q}(B_G)
is essentially the Eilenberg–Moore spectral sequence.
We briefly recall how it allows us to relate the rational cohomologies of G and of B_G, for connected G.
The algebra \operatorname {H}^\bullet (G,\mathbb {Q}) is a connected graded Hopf algebra of finite dimension over \mathbb {Q}.
If P^\bullet (G) is the graded module of its primitive elements, then we have
H^\bullet (G,\mathbb {Q}) = \bigwedge P^\bullet (G)
and the generators of P^\bullet (G) are of odd degree.
The simplicial algebra (E_1^{p\bullet })_{p\geqslant 0} is
E_1^{p\bullet } = \bigwedge \big (P^\bullet (G)^{\Delta _p}/P^\bullet (G)\big )
which is the exterior algebra of the suspension of the constant cosimplicial module P^\bullet (G);
we thus have (by Quillen [Q1968]) that E_2^{p\bullet }=\operatorname {Sym}^p(P^\bullet (G)).
The pages E_2^{pq} are only zero for p+q even;
we thus have that E_2^{pq}=E_\infty ^{pq}, and, for a suitable filtration, we canonically have that
\operatorname {Gr}\operatorname {H}^\bullet (B_G,\mathbb {Q}) \simeq \operatorname {Sym}^\bullet (P^\bullet (G)[-1])
and non-canonically that
\operatorname {H}^\bullet (B_G,\mathbb {Q}) \simeq \operatorname {Sym}^\bullet (P^\bullet (G)[-1]).
2180hodge-theory-iii-6.1.6hodge-theory-iii-6.1.6.xml6.1.6hodge-theory-iii-6.1
Let G be a (complex) linear algebraic group.
If T is a maximal torus of G, with Weyl group W, then
253Equationhodge-theory-iii-6.1.6.1hodge-theory-iii-6.1.6.1.xml6.1.6.1hodge-theory-iii-6.1.6 \operatorname {H}^\bullet (B_G,\mathbb {Q}) \xrightarrow {\sim } \operatorname {H}^\bullet (B_T,\mathbb {Q})^W. \tag{6.1.6.1}
If T is a torus with character group X(T), then
255Equationhodge-theory-iii-6.1.6.2hodge-theory-iii-6.1.6.2.xml6.1.6.2hodge-theory-iii-6.1.6 \operatorname {H}^\bullet (T,\mathbb {Z}) \simeq \bigwedge ^\bullet X(T) \tag{6.1.6.2}
(an isomorphism of graded Hopf algebras).
We will only use (a) in the following weaker form:
(The splitting principle).
The map \operatorname {H}^\bullet (B_G,\mathbb {Q})\to \operatorname {H}^\bullet (B_T,\mathbb {Q}) is injective.
For completion, we recall a proof of (a').
If B is a Borel subgroup of G, then the bundle U_G/B on B_G is a fibre in flag spaces.
By [2.1 and 2.6.3, D1968], which is better explained in [Proposition 3.1, G1970], or by [B1956], the Leray spectral sequence of U_G/B\to B_G degenerates to rational cohomology.
We thus have that \operatorname {H}^\bullet (B_G,\mathbb {Q})\hookrightarrow \operatorname {H}^\bullet (U_G/B,\mathbb {Q}).
We conclude by noting that U_G/B\sim B_B\sim B_T.
2752hodge-theory-iii-6.2hodge-theory-iii-6.2.xmlConstruction of hypercovers6.2hodge-theory-iii-62234hodge-theory-iii-6.2.1hodge-theory-iii-6.2.1.xml6.2.1hodge-theory-iii-6.2
Let X_\bullet be a simplicial set.
Denote by D(\Delta _n,\Delta _M) the set of increasing surjective maps from \Delta _n to \Delta _m (degeneracy operators), and set
2232Equationhodge-theory-iii-6.2.1.1hodge-theory-iii-6.2.1.1.xml6.2.1.1hodge-theory-iii-6.2.1 N(X_n) = X_n\setminus \bigcup _{{s\in D(\Delta _n,\Delta _{n-1})}}s(X_{n-1}). \tag{6.2.1.1}
Recall that, for all n, the map
2233Equationhodge-theory-iii-6.2.1.2hodge-theory-iii-6.2.1.2.xml6.2.1.2hodge-theory-iii-6.2.1 \coprod s \colon \coprod _{\mathclap {\substack {m\leqslant n,\\s\in D(\Delta _n,\Delta _m)}}} N(X_m) \to X_n \tag{6.2.1.2}
is bijective.
2235Definitionhodge-theory-iii-6.2.2hodge-theory-iii-6.2.2.xml6.2.2hodge-theory-iii-6.2
We say that a simplicial topological space is s-split if the maps in are homeomorphisms.
Let X_\bullet be a k-truncated simplicial set.
For n\leqslant k, we again define N(X_n) by , and then is a bijection.
We say that a k-truncated simplicial topological space is s-split if is a homeomorphism for n\leqslant k.
2237hodge-theory-iii-6.2.3hodge-theory-iii-6.2.3.xml6.2.3hodge-theory-iii-6.2
For an s-split (n+1)-truncated topological simplicial space X augmented over S, let \alpha (X) be the triple consisting of \operatorname {sq}_n(X), NX_{n+1}, and the evident map from NX_{n+1} to (\operatorname {cosq}\operatorname {sq}_n X)_{n+1}.
This triple \alpha (X)=(Y,N,\beta ) satisfies the following:
2236Propertyhodge-theory-iii-6.2.starhodge-theory-iii-6.2.star.xml*hodge-theory-iii-6.2.3(*) Y is an s-split n-truncated simplicial topological space augmented over S, and \beta is a continuous map from N to (\operatorname {cosq} Y)_{n+1}.
2246Propositionhodge-theory-iii-6.2.4hodge-theory-iii-6.2.4.xml6.2.4hodge-theory-iii-6.2
Let (Y,N,\beta ) satisfy above.
Up to unique isomorphism, there exists exactly one s-split (n+1)-truncated topological space X augmented over S such that \alpha (X)\simeq (Y,N,\beta ).
It is equivalent to give either f\colon X\to Z or:
a morphism f'\colon Y\to \operatorname {sq}_n(Z); and
a morphism f''\colon N\to Z_{n+1}, such that the diagram
\begin {CD} N @>\beta >> (\operatorname {cosq} Y)_{n+1} \\@V{f''}VV @VV{f'}V \\Z_{n+1} @>>> (\operatorname {cosq}\operatorname {sq}_n Z)_{n+1} \end {CD}
commutes.
2244Proof#177unstable-177.xmlhodge-theory-iii-6.2.4
[5.1.3, SD].
This proposition also applies to simplicial objects in other categories \mathscr {C} apart from that of topological spaces; it suffices that \mathscr {C} satisfy the following:
2245hodge-theory-iii-6.2.4.1hodge-theory-iii-6.2.4.1.xml6.2.4.1hodge-theory-iii-6.2.4
Finite projective limits exist in \mathbb {C}.
Finite projective sums exist in \mathbb {C}, and they are disjoint and universal.
2247hodge-theory-iii-6.2.5hodge-theory-iii-6.2.5.xml6.2.5hodge-theory-iii-6.2 allows us to construct, by induction, proper hypercovers of S.
We take f_0\colon X_0\to S, proper and surjective.
Then \{X_0\} is a 0-truncated proper hypercover of S (), and it is s-split.
We take f_1\colon N_1\to \operatorname {cosq}(\{X_0\})_1, i.e. f_1\colon N_1\to X_0\times _S X_0.
Applying , we associate to f_1 the s-split 1-truncated augmented simplicial topological space
{}_1X_\bullet = \left ( N_1{\textstyle \coprod } X_0 \xleftarrow [\to ]{\to } X_0 \to S \right ).
Suppose f_1 to be chosen such that
f'_1 \colon N_1{\textstyle \coprod } X_0\to X_0\times _S X_0
is proper and surjective (for example, if f_1 is proper and surjective).
Then {}_1X_\bullet is an s-split 1-truncated proper hypercover of S.
Assume that we have already constructed an s-split k-truncated proper hypercover {}_kX_\bullet \to S.
We take f_{k+1}\colon N_{k+1}\to (\operatorname {cosq}({}_kX_\bullet ))_{k+1} and, applying , we associate to f_{k+1} an s-split (k+1)-truncated augmented semi-simplicial space {}_{k+1}X_\bullet .
Suppose that f_{k+1} is such that
f'_{k+1} \colon {}_{k+1}X_{k+1} \to \operatorname {cosq}({}_kX_\bullet )_{k+1}
is proper and surjective (for example, if f_{k+1} is proper and surjective).
Then {}_{k+1}X_\bullet is an s-split (k+1)-truncated proper hypercover of S.
The {}_kX_\bullet thus constructed are the successive skeletons of an s-split proper hypercover of S.
2248hodge-theory-iii-6.2.6hodge-theory-iii-6.2.6.xml6.2.6hodge-theory-iii-6.2
We say that a simplicial scheme X_\bullet over \mathbb {C} is smooth if the X_n are smooth;
it is said to be proper if the X_n are compact.
A normal crossing divisor D_\bullet of X_\bullet , assumed to be smooth, is a family D_n\subset X_n of normal crossing divisors () such that the U_n=X_n\setminus D_n form a simplicial subscheme U_\bullet of X_\bullet .
This definition is justified by the following lemma.
2249Lemmahodge-theory-iii-6.2.7hodge-theory-iii-6.2.7.xml6.2.7hodge-theory-iii-6.2
If D_\bullet is a normal crossing divisor of X_\bullet , then the logarithmic de Rham complexes (\Omega _{X_n}^\bullet (\log D_n))_{n\geqslant 0}, endowed with the weight filtration (), form a filtered complex on X_\bullet .
1378Proof#178unstable-178.xmlhodge-theory-iii-6.2.7
This follows from (ii) of .
We denote the complex (\Omega _{X_n}^\bullet (\log D_n))_{n\geqslant 0} by \Omega _{X_\bullet }^\bullet (\log D_\bullet ).
2254hodge-theory-iii-6.2.8hodge-theory-iii-6.2.8.xml6.2.8hodge-theory-iii-6.2
Using , we can show that, for every separated scheme S over \mathbb {C}, there exists:
a simplicial scheme X_\bullet over \mathbb {C}, smooth and proper, that we can take to be s-split;
a normal crossing divisor D_\bullet of X_\bullet ; we set U_\bullet =X_\bullet \setminus D_\bullet ;
an augmentation a\colon U_\bullet \to S that realises U_\bullet ^\mathrm {an} as a proper hypercover of S^\mathrm {an}.
Furthermore, any two such systems are covered by a third, and a morphism u\colon S\to T can be covered by a morphism
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
X_\bullet
\ar [r]
& Y_\bullet
\\U_\bullet
\ar [r]
\ar [u,hook]
\ar [d,swap,"a"]
& V_\bullet
\ar [u,hook]
\ar [d,"b"]
\\S \ar [r,swap,"u"]
& T
\end {tikzcd}
of such systems (see [SD]).
2753hodge-theory-iii-6.3hodge-theory-iii-6.3.xmlRelative cohomology6.3hodge-theory-iii-62302hodge-theory-iii-6.3.1hodge-theory-iii-6.3.1.xml6.3.1hodge-theory-iii-6.3
The mapping cone construction for morphisms of simplicial sets works in the same way for simplicial objects in any category \mathscr {C} that has a final object e and finite sums.
For u\colon Y_\bullet \to X_\bullet , the cone C(u) satisfies
C(u)_n = X_n{\,\textstyle \coprod \,}\coprod _{i<n} Y_i{\,\textstyle \coprod \,}e.
We take \mathscr {C} to be:
the category of topological spaces and continuous maps, with final object e=P^t;
the category of pairs (X,F) where X is a topological space and F is an abelian sheaf on X, with an arrow (u,f)\colon (Y,F)\to (X,G) consisting of a continuous map u\colon Y\to X and a u-morphism () f\colon G\to F, with final object e=(P^t,0).
2305hodge-theory-iii-6.3.2hodge-theory-iii-6.3.2.xml6.3.2hodge-theory-iii-6.3
Let u\colon Y_\bullet \to X_\bullet be a morphism of topological simplicial spaces, with cone C(u).
Let F be an abelian sheaf on X_\bullet and G an abelian sheaf on Y_\bullet , and let f\colon G\to F be a u-morphism.
The cone C(f) of f is an abelian sheaf on C(u), and we set
2303Equationhodge-theory-iii-6.3.2.1hodge-theory-iii-6.3.2.1.xml6.3.2.1hodge-theory-iii-6.3.2 \operatorname {H}^n(C(u),C(f)) = \operatorname {H}^n(X_\bullet \operatorname {mod} Y_\bullet , F\operatorname {mod} G). \tag{6.3.2.1}
These are the relative cohomology groups.
We can easily show that they fit into a long exact sequence
2304Equationhodge-theory-iii-6.3.2.2hodge-theory-iii-6.3.2.2.xml6.3.2.2hodge-theory-iii-6.3.2 \ldots \to \operatorname {H}^i(X_\bullet \operatorname {mod} Y_\bullet , F\operatorname {mod} G) \to \operatorname {H}^i(X_\bullet ,F) \to \operatorname {H}^i(Y_\bullet ,G) \to \ldots . \tag{6.3.2.2} 2307hodge-theory-iii-6.3.3hodge-theory-iii-6.3.3.xml6.3.3hodge-theory-iii-6.3
More generally, let L and K be bounded-below complexes of abelian sheaves on Y_\bullet and X_\bullet (respectively), and let f\colon K\to L be a u-morphism.
We thus obtain a complex C(f) on C(u).
We again define the hypercohomology
\operatorname {\mathbb {H}}^n(C(u),C(f)) = \operatorname {\mathbb {H}}^n(X_\bullet \operatorname {mod} Y_\bullet , K\operatorname {mod} L).
These groups appear in an exact sequence analogous to that of , coming, in the suitable derived category, from a distinguished triangle
2306Equationhodge-theory-iii-6.3.3.1hodge-theory-iii-6.3.3.1.xml6.3.3.1hodge-theory-iii-6.3.3
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
& \mathrm {R}\Gamma (Y_\bullet ,K)
\ar [dl,swap,"\partial "]
\\\mathrm {R}\Gamma (C(u),C(f))
\ar [rr]
&& \mathrm {R}\Gamma (X_\bullet ,L)
\ar [ul]
& (6.3.3.1)
\end {tikzcd}
2308hodge-theory-iii-6.3.4hodge-theory-iii-6.3.4.xml6.3.4hodge-theory-iii-6.3
The construction presented above is not the only one possible.
It has the inconvenience that, even if we start with true topological spaces X and Y (i.e. constant simplicial spaces), we are led to consider non-constant simplicial spaces.
2754hodge-theory-iii-6.4hodge-theory-iii-6.4.xmlMultisimplicial spaces6.4hodge-theory-iii-62325hodge-theory-iii-6.4.1hodge-theory-iii-6.4.1.xml6.4.1hodge-theory-iii-6.4
Let r\geqslant 0 be an integer.
An r-simplicial object Z_\bullet in a category \mathscr {C} is a contravariant functor from the r-fold product (\Delta )^r to \mathscr {C}.
The diagonal simplicial object \delta Z_\bullet is the composite functor (\Delta )\to (\Delta )^r\to \mathscr {C}.
2326hodge-theory-iii-6.4.2hodge-theory-iii-6.4.2.xml6.4.2hodge-theory-iii-6.4
As in , we define the topos of sheaves on an r-simplicial topological space.
Let \Gamma ^\bullet be the functor
F \mapsto (\Gamma (X_{n_1\ldots n_r},F^{n_1\ldots n_r}))
from sheaves on X_\bullet to r-cosimplicial sets.
For small r, we often prefer to write \Gamma ^{\bullet \ldots \bullet } (with r-many copies of \bullet ).
We have a co-augmentation \Gamma (X_\bullet ,F^\bullet )\to \Gamma ^\bullet (X_\bullet ,F^\bullet ).
The functors of cohomology with values in an abelian sheaf F are the derived functors of the "global sections" functor \Gamma .
They can be calculated by a procedure parallel to that of , i.e.
2014Equationhodge-theory-iii-6.4.2.1hodge-theory-iii-6.4.2.1.xml6.4.2.1hodge-theory-iii-6.4.2 \mathrm {R}\Gamma =\mathrm {s}\mathrm {R}\Gamma ^\bullet \colon \mathrm {D}^+(X_\bullet ) \to \mathrm {D}^+((\mathsf {Ab})). \tag{6.4.2.1}
A sheaf F on an r-simplicial topological space X_\bullet induces a sheaf \delta F on the diagonal simplicial space \delta X_\bullet .
It follows from the Cartier–Eilenberg–Zilber theorem that
2015Equationhodge-theory-iii-6.4.2.2hodge-theory-iii-6.4.2.2.xml6.4.2.2hodge-theory-iii-6.4.2 \mathrm {R}\Gamma (X_\bullet ,K) \xrightarrow {\sim } \mathrm {R}\Gamma (\delta X_\bullet ,\delta K). \tag{6.4.2.2} 2327hodge-theory-iii-6.4.3hodge-theory-iii-6.4.3.xml6.4.3hodge-theory-iii-6.4
We restrict to the case r=2.
A bisimplicial object 1-augmented over a simplicial object S_\bullet is a contravariant functor from (\Delta ^+)\times (\Delta ) to \mathscr {C} such that S_\bullet is the composite functor
(\Delta ) \xrightarrow {(\Delta _{-1},\bullet )} (\Delta ^+)\times (\Delta ) \to \mathscr {C}.
To denote a bisimplicial object 1-augmented over S_\bullet , with underlying bisimplicial object X_{\bullet \bullet }, we will use notation of the form
a\colon X_{\bullet \bullet } \to S_\bullet .
For n\geqslant 0, a\colon (X_{\bullet n})\to S_n is a simplicial object augmented over S_n.
If F is a sheaf on X_{\bullet \bullet }, then the (a_*(F|X_{\bullet n})\text { on }S_n)_{n\geqslant 0} form a sheaf on S_\bullet ;
we thus define a topos morphism
a \colon X_{\bullet \bullet }^\sim \to S_\bullet ^\sim .
We explain in [SD] that \mathrm {R} a_* can be calculated "component by component", i.e.
\mathrm {R} a_*K|S_n = \mathrm {R} a_*(K|X_{\bullet n}).
It thus follows that if, for each n, the morphism a\colon (X_{\bullet n})\to S_n is of cohomological descent, then a\colon X_{\bullet \bullet }\to S_\bullet is of cohomological descent: for every complex K\in \mathrm {D}^+(S_\bullet ) of abelian sheaves, we have
K \xrightarrow {\approx } \mathrm {R} a_*a^*K. 2328hodge-theory-iii-6.4.4hodge-theory-iii-6.4.4.xml6.4.4hodge-theory-iii-6.4
In [SD], we show that, for every separated simplicial scheme S_\bullet , there exists a bisimplicial scheme X_{\bullet \bullet } that is 1-augmented over S_\bullet , along with i\colon X_{\bullet \bullet }\hookrightarrow \bar {X}_{\bullet \bullet } such that:
The \bar {X}_{nm} are smooth and projective;
X_{nm} is the complement of a normal crossing divisor D_{nm} in \bar {X}_{nm}, and we can suppose the D_{nm} to be a union of smooth divisors.
For n\geqslant 0, X_{\bullet n} is a proper hypercover of S_n, and we can take it to be s-split.
The construction proceeds as in , but the induction is more complicated.
The claims of uniqueness () remain true, mutatis mutandis.
2775hodge-theory-iii-7hodge-theory-iii-7.xmlSupplements to §17hodge-theory-iii2773hodge-theory-iii-7.1hodge-theory-iii-7.1.xmlFiltered derived category7.1hodge-theory-iii-7
This section completes .
2350hodge-theory-iii-7.1.1hodge-theory-iii-7.1.1.xml7.1.1hodge-theory-iii-7.1
Let \mathscr {A} be an abelian category.
We set:
\mathrm {F}\mathscr {A} (resp. \mathrm {F}_2\mathscr {A})
= the category of filtered (resp. bifiltered) objects with finite filtration(s) of \mathscr {A}.
\mathrm {K}^+\mathrm {F}\mathscr {A} (resp. \mathrm {K}^+\mathrm {F}_2\mathscr {A})
= the category of filtered (resp. bifiltered) bounded-below complexes of objects of \mathscr {A}, up to homotopy that respects the filtration(s).
\mathrm {D}^+\mathrm {F}\mathscr {A} (resp. \mathrm {D}^+\mathrm {F}_2\mathscr {A})
= the triangulated category induced from \mathrm {K}^+\mathrm {F}\mathscr {A} (resp. from \mathrm {K}^+\mathrm {F}_2\mathscr {A}) by inverting the filtered (resp. bifiltered) quasi-isomorphisms ();
this is the derived filtered category (resp. derived bifiltered category).
2351hodge-theory-iii-7.1.2hodge-theory-iii-7.1.2.xml7.1.2hodge-theory-iii-7.1
A filtered quasi-isomorphism u\colon (K,F)\to (K',F') induces an isomorphism of spectral sequences u\colon E_\bullet ^{\bullet \bullet }(K,F)\to E_\bullet ^{\bullet \bullet }(K',F').
An object K of \mathrm {D}^+\mathrm {F}\mathscr {A} thus defines a spectral sequence E_\bullet ^{\bullet \bullet }(K).
Similarly, an object L of \mathrm {D}^+\mathrm {F}_2\mathscr {A} defines an accumulation of spectral sequences of the type considered in .
2352hodge-theory-iii-7.1.3hodge-theory-iii-7.1.3.xml7.1.3hodge-theory-iii-7.1
Let T be a left exact functor from \mathscr {A} to an abelian category \mathscr {B}.
Suppose that every object of \mathscr {A} injects into an injective object.
The functor T can then be "derived" to give the functors
1063Equationhodge-theory-iii-7.1.3.1hodge-theory-iii-7.1.3.1.xml7.1.3.1hodge-theory-iii-7.1.3 \mathrm {R}\colon \mathrm {D}^+(\mathscr {A}) \to \mathrm {D}^+(\mathscr {B}) \tag{7.1.3.1}
1064Equationhodge-theory-iii-7.1.3.2hodge-theory-iii-7.1.3.2.xml7.1.3.2hodge-theory-iii-7.1.3 \mathrm {R}\colon \mathrm {D}^+\mathrm {F}(\mathscr {A}) \to \mathrm {D}^+\mathrm {F}(\mathscr {B}) \tag{7.1.3.2}
1065Equationhodge-theory-iii-7.1.3.3hodge-theory-iii-7.1.3.3.xml7.1.3.3hodge-theory-iii-7.1.3 \mathrm {R}\colon \mathrm {D}^+\mathrm {F}_2(\mathscr {A}) \to \mathrm {D}^+\mathrm {F}_2(\mathscr {B}). \tag{7.1.3.3}
They can be calculated as follows: if K' is a T-acyclic resolution (resp. a filtered resolution, resp. a bifiltered resolution) of K ( and ), then \mathrm {R} T(K)=T(K').
The hypercohomology spectral sequence (for T) of K\in \operatorname {Ob}\mathrm {D}^+\mathrm {F}(\mathscr {A}) is the spectral sequence of \mathrm {R} TK\in \operatorname {Ob}\mathrm {D}^+\mathrm {F}(\mathscr {B}) (cf. ).
2353hodge-theory-iii-7.1.4hodge-theory-iii-7.1.4.xml7.1.4hodge-theory-iii-7.1
We will need more precise results for the functors \mathrm {R} a_*, where a\colon X_\bullet \to S is an augmentation of a simplicial topological space.
The case S=P^t, where \mathrm {R} a_*=\mathrm {R}\Gamma will suffice.
We reuse the notation of , and recall :
\mathrm {R} a_* = \mathrm {s}\mathrm {R} {a_\bullet }_*.
For every complex K\in \operatorname {Ob}\mathrm {C}^+(S_\bullet ), the simple complex \mathrm {s} K is endowed with a natural filtration L ().
A quasi-isomorphism u\colon K'\xrightarrow {\sim } K'' induces a filtered quasi-isomorphism u\colon (\mathrm {s} K',L)\xrightarrow {\sim }(\mathrm {s} K'',L).
Then \mathrm {s} factors as
\mathrm {s}\colon \mathrm {D}^+(S_\bullet ) \to \mathrm {D}^+\mathrm {F}(S)
and \mathrm {R} a_* factors as
\mathrm {R} a_*\colon \mathrm {D}^+(X_\bullet ) \to \mathrm {D}^+\mathrm {F}(S).
The spectral sequence of the filtered complex (\mathrm {R} a_* K,L)\in \mathrm {D}^+\mathrm {F}(S) is exactly .
2356hodge-theory-iii-7.1.5hodge-theory-iii-7.1.5.xml7.1.5hodge-theory-iii-7.1
If K is filtered (resp. bifiltered), then \mathrm {R}{a_\bullet }_*K is filtered (resp. bifiltered): we have
2354Equationhodge-theory-iii-7.1.5.1hodge-theory-iii-7.1.5.1.xml7.1.5.1hodge-theory-iii-7.1.5 \mathrm {R}{a_\bullet }_* \colon \mathrm {D}^+\mathrm {F}(X_\bullet ) \to \mathrm {D}^+\mathrm {F}(S_\bullet ) \tag{7.1.5.1}
2355Equationhodge-theory-iii-7.1.5.2hodge-theory-iii-7.1.5.2.xml7.1.5.2hodge-theory-iii-7.1.5 \mathrm {R}{a_\bullet }_* \colon \mathrm {D}^+\mathrm {F}_2(X_\bullet ) \to \mathrm {D}^+\mathrm {F}_2(S_\bullet ). \tag{7.1.5.2} 2362hodge-theory-iii-7.1.6hodge-theory-iii-7.1.6.xml7.1.6hodge-theory-iii-7.1
Let K be a complex of cosimplicial sheaves on S, endowed with an increasing filtration W.
We define the diagonal filtration \delta (W,L) of W and L to be the increasing filtration of \mathrm {s} K given by
2357Equationhodge-theory-iii-7.1.6.1hodge-theory-iii-7.1.6.1.xml7.1.6.1hodge-theory-iii-7.1.6 \begin {aligned} \delta (W,L)_n(\mathrm {s} K) &= \bigoplus _{p,q} W_{n+p}(K^{p,q}) \\&= \sum _p \mathrm {s}(W_{n+p}(K)) \cap L^p(\mathrm {s} K). \end {aligned} \tag{7.1.6.1}
We have
2358Equationhodge-theory-iii-7.1.6.2hodge-theory-iii-7.1.6.2.xml7.1.6.2hodge-theory-iii-7.1.6 \operatorname {Gr}_n^{\delta (W,L)}(\mathrm {s} K) \simeq \bigoplus _p \operatorname {Gr}_{n+p}^W(K^{\bullet p})[-p]. \tag{7.1.6.2}
The functor (K,W)\mapsto (\mathrm {s} K,\delta (W,L)) sends filtered quasi-isomorphisms to filtered quasi-isomorphisms, and defines
2359Equationhodge-theory-iii-7.1.6.3hodge-theory-iii-7.1.6.3.xml7.1.6.3hodge-theory-iii-7.1.6 \begin {aligned} (s,\delta ) \colon \mathrm {D}^+\mathrm {F}(S_\bullet ) &\to \mathrm {D}^+\mathrm {F}(S) \\(K,W) &\mapsto (\mathrm {s} K,\delta (W,L)) \end {aligned} \tag{7.1.6.3}
whence, by composition with ,
2360Equationhodge-theory-iii-7.1.6.4hodge-theory-iii-7.1.6.4.xml7.1.6.4hodge-theory-iii-7.1.6 \begin {aligned} (\mathrm {R}\Gamma ,\delta (-,L)) \colon \mathrm {D}^+\mathrm {F}(X_\bullet ) &\to \mathrm {D}^+\mathrm {F}(S) \\(K,W) &\mapsto (\mathrm {R}\Gamma K,\delta (W,L)). \end {aligned} \tag{7.1.6.4}
From , we see that
2361Equationhodge-theory-iii-7.1.6.5hodge-theory-iii-7.1.6.5.xml7.1.6.5hodge-theory-iii-7.1.6 \operatorname {Gr}_n^{\delta (W,L)} = \bigoplus _p \mathrm {R}{a_p}_*(\operatorname {Gr}_{n+p}^W K)[-p]. \tag{7.1.6.5} 2366hodge-theory-iii-7.1.7hodge-theory-iii-7.1.7.xml7.1.7hodge-theory-iii-7.1
If (K,W,F) is a bifiltered complex of cosimplicial sheaves, then \mathrm {s} K is endowed with the three filtrations W, F, and L, and defines different bifiltered complexes.
For example, for increasing W, the functor (K,W,F)\mapsto (K,\delta (W,L),F) sends bifiltered quasi-isomorphisms to bifiltered quasi-isomorphisms, and thus defines
2363Equationhodge-theory-iii-7.1.7.1hodge-theory-iii-7.1.7.1.xml7.1.7.1hodge-theory-iii-7.1.7 \begin {aligned} \mathrm {D}^+\mathrm {F}_2(S_\bullet ) &\to \mathrm {D}^+\mathrm {F}_2(S) \\(K,W,F) &\mapsto (K,\delta (W,L),F). \end {aligned} \tag{7.1.7.1}
By composition with , we thus obtain
2364Equationhodge-theory-iii-7.1.7.2hodge-theory-iii-7.1.7.2.xml7.1.7.2hodge-theory-iii-7.1.7 \begin {aligned} \mathrm {D}^+\mathrm {F}_2(X_\bullet ) &\to \mathrm {D}^+\mathrm {F}_2(S) \\(K,W,F) &\mapsto (\mathrm {R}\Gamma K,\delta (W,L),F). \end {aligned} \tag{7.1.7.2}
and we have that
2365Equationhodge-theory-iii-7.1.7.3hodge-theory-iii-7.1.7.3.xml7.1.7.3hodge-theory-iii-7.1.7 \operatorname {Gr}_n^{\delta (W,L)}(\mathrm {R}\Gamma K,F) = \bigoplus _p \mathrm {R}{a_p}_*(\operatorname {Gr}_{n+p}^W K,F)[-p] \tag{7.1.7.3}
in \mathrm {D}^+\mathrm {F}(S).
2774hodge-theory-iii-7.2hodge-theory-iii-7.2.xmlSupplements to the two filtrations lemma7.2hodge-theory-iii-7
In this section, we give a new proof of the two filtrations lemma () and some supplements.
961hodge-theory-iii-7.2.1hodge-theory-iii-7.2.1.xml7.2.1hodge-theory-iii-7.2
Let (K,F,W) be a bounded below bifiltered complex with objects in an abelian category \mathscr {A}, with F assumed to be biregular.
We say that (K,F,W) is F-splitable if the filtered complex (K,W) can be written as a sum of filtered complexes
(K,W) = \bigoplus _{n\in \mathbb {Z}} (K_n,W_n)
with
F^nK = \bigoplus _{n'\geqslant n} K_{n'}.
Let r_0\geqslant 0 be an integer or +\infty .
The following condition was considered in and :
962Conditionhodge-theory-iii-7.2.2hodge-theory-iii-7.2.2.xml7.2.2hodge-theory-iii-7.2
For every non-negative integer r<r_0, the differentials d_r of the graded complex E_r(K,W) are strictly compatible with the recurrent filtration defined by F.
963hodge-theory-iii-7.2.3hodge-theory-iii-7.2.3.xml7.2.3hodge-theory-iii-7.2
It is clear that, if (K,F,W) is F-splitable, then is satisfied for r_0=\infty .
Conversely, it seems that if is satisfied for r_0=\infty , then everything is as if the functor \operatorname {Gr}_F were exact.
For example, we will show that the \operatorname {Gr}_F of the spectral sequence E(K,W) can then be identified with the spectral sequence E(\operatorname {Gr}_FK,W), and that the spectral sequence E(K,F) degenerates (E_1=E_\infty ).
964hodge-theory-iii-7.2.4hodge-theory-iii-7.2.4.xml7.2.4hodge-theory-iii-7.2
We immediately deduce from the definition () that the first direct filtration of E_r(K,W) is the filtration F_d of E_r(K,W) by the images
F_d^p(E_r(K,W)) = \operatorname {Im} \big (E_r(F^pK,W) \to E_r(K,W)\big ).
Dually, the second direct filtration () of E_r(K,W) is the filtration F_{d^*} of E_r(K,W) by the kernels
F_{d^*}^p(E_r(K,W)) = \operatorname {Ker} \big (E_r(K,W) \to E_r(K/F^pK,W)\big ).
The recurrent filtration F_{\mathrm {rec}} of E_r(K,W) is intermediary between these two filtrations ((iii) of ).
969Propositionhodge-theory-iii-7.2.5hodge-theory-iii-7.2.5.xml7.2.5hodge-theory-iii-7.2
Suppose that (K,F,W) satisfies for some r_0\geqslant 0.
Then
(F^aK/F^bK,F,W) also satisfies for r_0.
For r\leqslant r_0, the sequence
0 \to E_r(F^pK,W) \to E_r(K,W) \to E_r(K/F^pK,W) \to 0
is exact;
for r=r_0+1, the sequence
E_r(F^pK,W) \to E_r(K,W) \to E_r(K/F^pK,W)
is exact.
In particular, for r\leqslant r_0+1, the two direct filtrations and the recurrent filtration of E_r(K,W) agree.
968Proof#176unstable-176.xmlhodge-theory-iii-7.2.5
Fix
3014hodge-theory-iii-5hodge-theory-iii-5.xmlHodge Theory III › Cohomological descent5hodge-theory-iii2768hodge-theory-iii-5.1hodge-theory-iii-5.1.xmlSimplicial topological spaces5.1hodge-theory-iii-5
This section starts with some reminders for which we can refer to the homotopical seminar of Strasbourg, 1963/64.
1937hodge-theory-iii-5.1.1hodge-theory-iii-5.1.1.xml5.1.1hodge-theory-iii-5.1
We will us the following notation, where n,k\geqslant -1 are integers.
\mathscr {A}^\circ
= the opposite category of a category \mathscr {A}
\underline {\operatorname {Hom}}(\mathscr {A},\mathscr {B})
= the category of functors from \mathscr {A} to \mathscr {B}.
\Delta _n
= the finite totally ordered set [0,n].
\delta _i\colon \Delta _n\to \Delta _{n+1}
= the increasing injection such that i\not \in \delta _i(\Delta _n) (for 0\leqslant i\leqslant n+1).
s_i\colon \Delta _{n+1}\to \Delta _n
= the increasing surjection such that s_i(i)=s_i(i+1) (for 0\leqslant i\leqslant n).
\varepsilon \colon \Delta _{-1}\to \Delta _n
= the unique map from \Delta _{-1} to \Delta _n.
(\Delta ^+)
= the category whose objects are the \Delta _n (for n\geqslant -1) and whose morphisms are the increasing maps between the \Delta _n.
(\Delta )
= the full subcategory of (\Delta ^+) whose objects are the \Delta _n (for n\geqslant 0).
(\Delta ^+)_k
= the full subcategory of (\Delta ^+) whose objects are the \Delta _n (for k\geqslant n).
(\Delta )_k
= the full subcategory of (\Delta ^+) whose objects are the \Delta _n (for k\geqslant n\geqslant 0).
For any category \mathscr {C}, we define a simplicial object (resp. k-truncated simplicial object) of \mathscr {C} to be an object of \underline {\operatorname {Hom}}((\Delta )^0,\mathscr {C}) (resp. of \underline {\operatorname {Hom}}((\Delta )^0_k,\mathscr {C})).
Similarly, a cosimplicial object (resp. k-truncated cosimplicial object) is an object of \underline {\operatorname {Hom}}((\Delta ),\mathscr {C}) (resp. of \underline {\operatorname {Hom}}((\Delta )_k,\mathscr {C})).
For k\geqslant -1, the k-skeleton functor is the restriction functor
\operatorname {sq}_k\colon \underline {\operatorname {Hom}}((\Delta )^0,\mathscr {C}) \to \underline {\operatorname {Hom}}((\Delta )_k^0,\mathscr {C})
and the k-coskeleton functor
\operatorname {cosq}_k\colon \underline {\operatorname {Hom}}((\Delta )_k^0,\mathscr {C}) \to \underline {\operatorname {Hom}}((\Delta )^0,\mathscr {C}).
is the right adjoint to \operatorname {sq}_k.
Let Y be a simplicial object of \mathscr {C}.
We also call the functor
\operatorname {sq}_k\colon \underline {\operatorname {Hom}}((\Delta )^0,\mathscr {C})/Y \to \underline {\operatorname {Hom}}((\Delta )_k^0,\mathscr {C})/\operatorname {sq}_k(Y)
the skeleton, and its right adjoint
\operatorname {cosq}_k^Y\colon \underline {\operatorname {Hom}}((\Delta )_k^0,\mathscr {C})/\operatorname {sq}_k(Y) \to \underline {\operatorname {Hom}}((\Delta )^0,\mathscr {C})/Y
is called the coskeleton with respect to Y.
The coskeleton functors exist if finite projective limits exist in \mathscr {C};
the relative coskeleton functors exist if fibred products do;
we have
\operatorname {cosq}_k^Y(X) = \operatorname {cosq}_k(X) \times _{\operatorname {cosq}_k\operatorname {sq}_k(Y)}Y. 1938hodge-theory-iii-5.1.2hodge-theory-iii-5.1.2.xml5.1.2hodge-theory-iii-5.1
If X_\bullet \colon (\Delta )^0\to \mathscr {C} is a simplicial object of \mathscr {C}, then we set
X_n = X_\bullet (\Delta _n)
\delta _i = X_\bullet (\delta _i\colon \Delta _n\to \Delta _{n+1})\colon X_{n+1}\to X_n
s_i = X_\bullet (s_i\colon \Delta _{n+1}\to \Delta _n)\colon X_n\to X_{n+1}.
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
X_\bullet = \Big (\ldots \quad
X_2
\ar [r,"\delta _1" description]
\ar [r,shift left=6,"\delta _0" description]
\ar [r,shift right=6,"\delta _2" description]
& X_1
\ar [r,shift left=3,"\delta _0" description]
\ar [r,shift right=3,"\delta _1" description]
\ar [l,shift right=3,"s_0" description]
\ar [l,shift left=3,"s_1" description]
& X_0\Big )
\ar [l,"s_0" description]
\end {tikzcd}1939hodge-theory-iii-5.1.3hodge-theory-iii-5.1.3.xml5.1.3hodge-theory-iii-5.1
Let S\in \operatorname {Ob}\mathscr {C}.
The constant simplicial object S_\bullet is the simplicial object for which S_n=S and \delta _i=s_i=\mathrm {Id}_S.
A simplicial (resp. k-truncated simplicial) object of \mathscr {C} augmented over S is a morphism a\colon X_\bullet \to S_\bullet (resp. a\colon X_\bullet \to \operatorname {sq}_k(S_\bullet )).
We identify such a simplicial object (resp. k-truncated simplicial object) augmented over S with the object X_\bullet ^+ of \underline {\operatorname {Hom}}((\Delta ^+)^0,\mathscr {C}) (resp. of \underline {\operatorname {Hom}}((\Delta ^+)_k^0,\mathscr {C})) such that X_\bullet ^+(\Delta _{-1})=S.
We have that a_n=X_\bullet ^+(\varepsilon \colon \Delta _{-1}\to \Delta _n).
These objects will be denoted by notation of the form a\colon X_\bullet \to S.
The relative coskeletons will be mostly used in this setting, and denoted \operatorname {cosq}_k^S or simply \operatorname {cosq}_k.
1940hodge-theory-iii-5.1.4hodge-theory-iii-5.1.4.xml5.1.4hodge-theory-iii-5.1
The coskeleton of the augmented 0-truncated simplicial object a_0\colon X\to S is the simplicial object augmented over S whose components are the cartesian powers of X in \mathscr {C}/S, i.e.
\operatorname {cosq}_0(X\to S) = \big ( ((X/S)^{\Delta _n})_{n\geqslant 0} \to S \big ). 1942hodge-theory-iii-5.1.5hodge-theory-iii-5.1.5.xml5.1.5hodge-theory-iii-5.1
Let u\colon X\to Y be a continuous map between topological spaces, and F a sheaf on X and G a sheaf on Y.
The set \operatorname {Hom}_u(G,F) of u-morphisms from G to F is the set \operatorname {Hom}(u^*G,F)\cong \operatorname {Hom}(G,u_*F).
The data of a u-morphism f from G to F consists of the data of, for each pair of opens (U\subset X,V\subset Y) such that u(U)\subset V, a map f_{UV}\colon G(V)\to F(U);
these maps must satisfy the condition
1941Conditionhodge-theory-iii-5.1.5.starhodge-theory-iii-5.1.5.star.xml*hodge-theory-iii-5.1.5
For U'\subset U\subset X and V'\subset V\subset Y such that u(U)\subset V and u(U')\subset V', the diagram
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
G(V)
\ar [r]
\ar [d]
& G(V')
\ar [d]
\\F(U)
\ar [r]
& F(U')
\end {tikzcd}
commutes.
1946hodge-theory-iii-5.1.6hodge-theory-iii-5.1.6.xml5.1.6hodge-theory-iii-5.1
A simplicial topological space is a simplicial object of the category whose objects are topological spaces and whose morphisms are continuous maps.
A sheaf F^\bullet on a simplicial topological space X_\bullet consists of
a family of sheaves F^n on the X_n;
for all f\colon \Delta _n\to \Delta _m, an X_\bullet (f)-morphism F^\bullet (f) from F^n to F^m.
We further require that F^\bullet (f\circ g)=F^\bullet (f)\circ F^\bullet (g).
A morphism u from F^\bullet to G^\bullet is a family of morphisms u^n\colon F^n\to G^n such that, for all f\in \operatorname {Hom}_{(\Delta )}(\Delta _n,\Delta _m), we have that u^m\circ F^\bullet (f)=G^\bullet (f)\circ u^n.
1950hodge-theory-iii-5.1.7hodge-theory-iii-5.1.7.xml5.1.7hodge-theory-iii-5.1
For an open U of X_n, let F^\bullet (U)=F^n(U).
For f\colon \Delta _n\to \Delta _m, U\subset X_n, and V\subset X_m such that X_\bullet (f)(V)\subset (U), let F^\bullet (f,V,U)\colon F^\bullet (U)\to F^\bullet (V) be the map induced by F^\bullet (f).
We immediately see that the system of sets F^\bullet (U) (indexed by n\geqslant 0 and U\subset X_n) and of maps F^\bullet (f,V,U) uniquely determines F^\bullet (cf. ).
For such a system to come from a sheaf on X_\bullet , it is necessary and sufficient that:
F^\bullet (fg,U,W)=F^\bullet (f,U,V)F^\bullet (g,V,W) whenever this makes sense;
for any n, the F^\bullet (U) for U\subset X_n form a sheaf on X_n.
1951hodge-theory-iii-5.1.8hodge-theory-iii-5.1.8.xml5.1.8hodge-theory-iii-5.1
This shows that sheaves on X_\bullet can be interpreted as sheaves on a suitable site, and, in particular, form a topos (X_\bullet )^\sim .
We freely use, for sheaves on X_\bullet , the modern terminology for sheaves on a site.
With this in mind, a sheaf F^\bullet of abelian groups (resp. of rings, ...) is a system of sheaves F^n of abelian groups (resp. of rings, ...) on the X_n along with the morphisms F^\bullet (f).
1960Exampleshodge-theory-iii-5.1.9hodge-theory-iii-5.1.9.xml5.1.9hodge-theory-iii-5.1
Let X_\bullet be a simplicial analytic space.
The structure sheaves \mathcal {O}_{X_n} form a sheaf of rings on X_\bullet .
Let X_\bullet be a simplicial analytic space augmented over S.
The sheaves \Omega _{X_n/S}^1 form a sheaf of \mathcal {O}-modules on X_\bullet .
Its i-th exterior power is the sheaf of \mathcal {O}-modules (\Omega _{X_n/S}^i)_{n\geqslant 0} denoted \Omega _{X_\bullet {/}S}^i.
The de Rham complexes (\Omega _{X_n/S}^\bullet )_{n\geqslant 0} form a complex of sheaves on X_\bullet .
Let F^\bullet be a sheaf of abelian groups on X_\bullet .
The canonical flasque Godement resolutions \mathscr {C}^\bullet (F^n) form a complex of sheaves on X_\bullet that is a resolution of F^\bullet .
Let S be a topological space.
Sheaves F^\bullet on the constant simplicial space S_\bullet can be identified with cosimplicial sheaves on S^\bullet .
In particular, an abelian sheaf F^\bullet on S_\bullet defines a chain complex (F^n,d=\sum _i(-1)^i\delta _i).
A complex K of abelian sheaves on S_\bullet defines a double complex (K^{n,m}) that we again denote by K (here m is the cosimplicial degree) and whose associated simple complex \mathrm {s}{K} is
1957Equationhodge-theory-iii-5.1.9.1hodge-theory-iii-5.1.9.1.xml5.1.9.1hodge-theory-iii-5.1.9 (\mathrm {s}{K})^n = \bigoplus _{p+q=n} K^{pq} \tag{5.1.9.1}
with differential given by
1958Equationhodge-theory-iii-5.1.9.2hodge-theory-iii-5.1.9.2.xml5.1.9.2hodge-theory-iii-5.1.9 \mathrm {d}(x^{pq}) = \mathrm {d}_K(x^{pq}) + (-1)^p\sum _i(-1)^i\delta _i x^{pq}. \tag{5.1.9.2}
We denote by L the second filtration of \mathrm {s}{K}, i.e.
1959Equationhodge-theory-iii-5.1.9.3hodge-theory-iii-5.1.9.3.xml5.1.9.3hodge-theory-iii-5.1.9 L^r(\mathrm {s}{K}) = \bigoplus _{q\geqslant r} K^{pq}. \tag{5.1.9.3}
1961hodge-theory-iii-5.1.10hodge-theory-iii-5.1.10.xml5.1.10hodge-theory-iii-5.1
Let u\colon X_\bullet \to Y_\bullet be a morphism of simplicial topological spaces, with components u_n\colon X_n\to Y_n.
If G (resp. F) is a sheaf on Y_\bullet (resp. on X_\bullet ), then (u_n^*G^n)_{n\geqslant 0} (resp. ({u_n}_*F^n)_{n\geqslant 0}) is a sheaf on X_\bullet (resp. on Y_\bullet );
we denote it by u^*G (resp. by u_*F).
The functors u^* and u_* are adjoint to one another;
they are the inverse image and direct image morphisms of topos morphism u\colon (X_\bullet )^\sim \to (Y_\bullet )^\sim .
1963hodge-theory-iii-5.1.11hodge-theory-iii-5.1.11.xml5.1.11hodge-theory-iii-5.1
Let a\colon X_\bullet \to S be an augmented simplicial topological space.
If F is a sheaf on S, then a^*F=(a_n^*F)_{n\geqslant 0} "is" a sheaf on X_\bullet .
The functor a^* has a right adjoint
1962Equationhodge-theory-iii-5.1.11.1hodge-theory-iii-5.1.11.1.xml5.1.11.1hodge-theory-iii-5.1.11 a_*\colon F^\bullet \mapsto \operatorname {Ker}\big ( {a_0}_* F^0 \,\overset {\delta _0^*}{\underset {\delta _1^*}{\rightrightarrows }}\, {a_1}_* F^1 \big ). \tag{5.1.11.1}
The functors a^* and a_* are the inverse image and direct image morphisms of topos morphism a\colon (X_\bullet )^\sim \to (S)^\sim .
1966hodge-theory-iii-5.1.12hodge-theory-iii-5.1.12.xml5.1.12hodge-theory-iii-5.1
Let S_\bullet be the constant simplicial space associated to S, and a_\bullet \colon X_\bullet \to S_\bullet the morphism defined by a.
For an abelian sheaf F^\bullet on X_\bullet , we can identify {a_\bullet }_*F^\bullet with a cosimplicial sheaf on S ((IV) of ).
For a complex K of abelian sheaves on X_\bullet , if K^{pq} is the degree p component of the restriction of K to X_q, then the components of the simple complex associated to the double complex defined by {a_\bullet }_*K are
1964Equationhodge-theory-iii-5.1.12.1hodge-theory-iii-5.1.12.1.xml5.1.12.1hodge-theory-iii-5.1.12 (\mathrm {s}{a_\bullet }_*K)^n = \bigoplus _{p+q=n} {a_q}_* K^{pq}. \tag{5.1.12.1}
The spectral sequence defined by the filtration L () of \mathrm {s}{a_\bullet }_*K can be written as
1965Equationhodge-theory-iii-5.1.12.2hodge-theory-iii-5.1.12.2.xml5.1.12.2hodge-theory-iii-5.1.12 E_1^{pq} = \operatorname {H}^q({a_p}_*(K|X^p)) \Rightarrow \operatorname {H}^{p+q}(\mathrm {s}{a_\bullet }_*K). \tag{5.1.12.2} 1968hodge-theory-iii-5.1.13hodge-theory-iii-5.1.13.xml5.1.13hodge-theory-iii-5.1
Let S=P^t (the topological space consisting of a single point).
We see that, for a sheaf F^\bullet on X_\bullet , the \Gamma (X_n,F^n) are the components of a cosimplicial set \Gamma ^\bullet (X_\bullet ,F^\bullet ).
The functor \Gamma (of global sections) is the functor
1967Equationhodge-theory-iii-5.1.13.1hodge-theory-iii-5.1.13.1.xml5.1.13.1hodge-theory-iii-5.1.13 \Gamma \colon F^\bullet \mapsto \operatorname {Ker}\big ( \Gamma (X_0,F^0) \rightrightarrows \Gamma (X_1,F^1) \big ). \tag{5.1.13.1}
If K is a complex of abelian sheaves, we denote by \Gamma ^\bullet (X_\bullet ,K) the chain complex whose components are the cosimplicial abelian groups \Gamma ^\bullet (X_\bullet ,K^n), and by \mathrm {s} \Gamma ^\bullet (X_\bullet ,K) the associated simple complex.
2769hodge-theory-iii-5.2hodge-theory-iii-5.2.xmlCohomology of simplicial topological spaces5.2hodge-theory-iii-52040hodge-theory-iii-5.2.1hodge-theory-iii-5.2.1.xml5.2.1hodge-theory-iii-5.2
To each simplicial topological space X_\bullet is associated a usual topological space |X_\bullet |, called its geometric realisation (see below).
In the particular case where the X_n are discrete, we recover the usual notion of geometric realisation of a simplicial set.
In [S1968], G. Segal defined the cohomology of X_\bullet with values in an abelian group A as \operatorname {H}^\bullet (|X_\bullet |,A).
Under suitable hypotheses, the filtration of |X_\bullet | by successive skeletons gives a spectral sequence
352Equationhodge-theory-iii-5.2.1.1hodge-theory-iii-5.2.1.1.xml5.2.1.1hodge-theory-iii-5.2.1 E_1^{pq} = \operatorname {H}^q(X_p,A) \Rightarrow \operatorname {H}^{p+q}(X_\bullet ,A). \tag{5.2.1.1}
One of the interests in using this definition is that it also applies, for example, to the definition of the groups K(X_\bullet ).
We will adopt another definition, which is better adapted to sheaf-theoretic techniques.
353#179unstable-179.xmlGeometric realisationhodge-theory-iii-5.2.1
For an integer n\geqslant 0, we denote by |\Delta _n| the simple in \mathbb {R}^{\Delta _n} whose vertices are the set of basis vectors.
We identify \Delta _n with the set of vertices of |\Delta _n|.
Every function f\colon \Delta _n\to \Delta _m extends by linearity to |f|\colon |\Delta _n|\to |\Delta _m|.
Let X_\bullet be a simplicial topological space.
Let
Y=\coprod _{n\geqslant 0}(X_n\times |\Delta _n|).
Let R be the finest equivalence relation on Y for which, for any f\colon \Delta _n\to \Delta _m in (\Delta ), any x_m\in X_m, and any a\in |\Delta _n|, we have
(x_m,|f|(a)) \equiv (f(x_m),a) \operatorname {mod} R.
The geometric realisation of X_\bullet is by definition
|X_\bullet | = Y/R. 2041Definitionhodge-theory-iii-5.2.2hodge-theory-iii-5.2.2.xml5.2.2hodge-theory-iii-5.2
Let X_\bullet be a simplicial topological space.
The functors \operatorname {H}^i(X_\bullet ,F^\bullet ) of cohomology with coefficients in the abelian sheaf F^\bullet on X_\bullet are the derived functors of the functor \Gamma ().
This definition is equivalent to the following, sometimes more convenient.
2042hodge-theory-iii-5.2.3hodge-theory-iii-5.2.3.xml5.2.3hodge-theory-iii-5.2
Let F^\bullet be an abelian sheaf on X_\bullet .
We can show (by example, using (III) of ) that F^\bullet always admits resolutions K on the right such that
\operatorname {H}^r(X_q,K^{p,q}) =0 \quad \text {for } p,q\geqslant 0 \text { and } r>0.
If K is such a resolution, then we canonically have that
1984Equationhodge-theory-iii-5.2.3.1hodge-theory-iii-5.2.3.1.xml5.2.3.1hodge-theory-iii-5.2.3 \operatorname {H}^n(X_\bullet ,F^\bullet ) \simeq \operatorname {H}^n(\mathrm {s}\Gamma ^\bullet (X_\bullet ,K)). \tag{5.2.3.1}
It is easy to show directly that the right-hand side is independent (up to unique isomorphism) of the choice of K;
for what follows, we are free to define \operatorname {H}^\bullet (X_\bullet ,F^\bullet ) by .
We can further show that the spectral sequence of the complex \mathrm {s}\Gamma ^\bullet (X_\bullet ,K), filtered by L ( and for S=P^t)
1985Equationhodge-theory-iii-5.2.3.2hodge-theory-iii-5.2.3.2.xml5.2.3.2hodge-theory-iii-5.2.3 E_1^{pq} = \operatorname {H}^q(X_p,F^p) \Rightarrow \operatorname {H}^{p+q}(X_\bullet ,F^\bullet ) \tag{5.2.3.2}
is independent (up to unique isomorphism) of the choice of K (cf. ).
2045hodge-theory-iii-5.2.4hodge-theory-iii-5.2.4.xml5.2.4hodge-theory-iii-5.2
We will need to make precise the above construction when passing to derived categories, and to give a relative variant of it.
Let a\colon X_\bullet \to S, let S_\bullet be the constant simplicial space defined by S with augmentation map \varepsilon \colon S_\bullet \to S, and let a_\bullet \colon X_\bullet \to S_\bullet be the map induced by a.
Then
2043Equationhodge-theory-iii-5.2.4.1hodge-theory-iii-5.2.4.1.xml5.2.4.1hodge-theory-iii-5.2.4 a_* = \varepsilon _* {a_\bullet }_*\colon (X_\bullet )^\sim \to (S)^\sim . \tag{5.2.4.1}
The derived version of this equation is
2044Equationhodge-theory-iii-5.2.4.2hodge-theory-iii-5.2.4.2.xml5.2.4.2hodge-theory-iii-5.2.4 \mathrm {R} a_* = \mathrm {R}\varepsilon _* \mathrm {R}{a_\bullet }_*\colon \mathrm {D}^+(X_\bullet ) \to \mathrm {D}^+(S) \to (S)^\sim . \tag{5.2.4.2}
where \mathrm {D}^+(X_\bullet ) is the bounded-below derived category of the category of abelian sheaves on X_\bullet .
We will calculate \mathrm {R}{a_\bullet }_* and \mathrm {R}\varepsilon _*.
2046hodge-theory-iii-5.2.5hodge-theory-iii-5.2.5.xml5.2.5hodge-theory-iii-5.2
Let u\colon X_\bullet \to Y_\bullet be a morphism of simplicial topological spaces.
The functor \mathrm {R} u_*\colon \mathrm {D}^+(X_\bullet )\to \mathrm {D}^+(Y_\bullet ) can be calculated "component by component": if K is a complex of abelian sheaves on X_\bullet , then to calculate \mathrm {R} u_*K we take a resolution K\xrightarrow {\sim } K' such that the components F of K' satisfy \mathrm {R}^i {u_p}_*(F^p)=0 for i>0, p\geqslant 0 (cf. (III) of );
then \mathrm {R} u_*K\approx u_*K'.
2048hodge-theory-iii-5.2.6hodge-theory-iii-5.2.6.xml5.2.6hodge-theory-iii-5.2
The functor \mathrm {s}\colon \operatorname {C}^+(S_\bullet )\to \operatorname {C}^+(S) sends acyclic complexes to acyclic complexes.
It thus trivially derives to
\mathrm {s}\colon \mathrm {D}^+(S_\bullet ) \to \mathrm {D}^+(S).
If F is an injective sheaf on S_\bullet , then the chain complex \mathrm {s} F is a resolution of \varepsilon _*F.
We thus obtain an isomorphism \mathrm {R}\varepsilon _*\xrightarrow {\sim }\mathrm {s}, and becomes
2047Equationhodge-theory-iii-5.2.6.1hodge-theory-iii-5.2.6.1.xml5.2.6.1hodge-theory-iii-5.2.6 \mathrm {R} a_* = \mathrm {s}\mathrm {R}{a_\bullet }_* \tag{5.2.6.1} 2049hodge-theory-iii-5.2.7hodge-theory-iii-5.2.7.xml5.2.7hodge-theory-iii-5.2
Combined with and specialised to the case where S=P^t, this equation proves .
In concrete terms, this implies that, to calculate \mathrm {R} a_*K, we can proceed in two steps:
We take a resolution K\xrightarrow {\sim } K' such that the components F of K' satisfy \mathrm {R}^i {a_p}_*(F^p)=0 for i>0.
The complex {a_\bullet }_*K'\in \mathrm {D}^+(S_\bullet ) (the derived category of the category of cosimplicial abelian sheaves on S) can be identified with \mathrm {R}{a_\bullet }_*K.
\mathrm {R} a_*K is the simple complex \mathrm {s}{a_\bullet }_*K' associated to the double complex {a_\bullet }_*K'.
The spectral sequence generalises to a spectral sequence
1990Equationhodge-theory-iii-5.2.7.1hodge-theory-iii-5.2.7.1.xml5.2.7.1hodge-theory-iii-5.2.7 E_1^{pq} = \mathrm {R}^q{a_p}_*(K|X_p) \Rightarrow \mathrm {R}^{p+q}a_*K \tag{5.2.7.1}
induced by .
2770hodge-theory-iii-5.3hodge-theory-iii-5.3.xmlCohomological descent5.3hodge-theory-iii-52105hodge-theory-iii-5.3.1hodge-theory-iii-5.3.1.xml5.3.1hodge-theory-iii-5.3
Let a\colon X_\bullet \to S be an augmented simplicial topological space.
For every sheaf F on S, we have a morphism
\varphi \colon F \to a_*a^*F
from the adjunction a^*\dashv a_*.
This morphism derives to a morphism of functors from \mathrm {D}^+(S) to \mathrm {D}^+(S), namely
2104Equationhodge-theory-iii-5.3.1.1hodge-theory-iii-5.3.1.1.xml5.3.1.1hodge-theory-iii-5.3.1 \varphi \colon \mathrm {Id} \to \mathrm {R} a_*a^*. \tag{5.3.1.1} 2106Definitionhodge-theory-iii-5.3.2hodge-theory-iii-5.3.2.xml5.3.2hodge-theory-iii-5.3
We say that a is of cohomological descent if, for every abelian sheaf F on S, we have
F\xrightarrow {\sim }\operatorname {Ker}({a_0}_*{a_0}^*F\to {a_1}_*{a_1}^*F)
and
\mathrm {R}^i a_*a^*F = 0 \quad \text {for }i>0.
This is equivalent to asking that be an isomorphism.
2107hodge-theory-iii-5.3.3hodge-theory-iii-5.3.3.xml5.3.3hodge-theory-iii-5.3
If a is of cohomological descent, then, for K\in \operatorname {Ob}\mathrm {D}^+(S), the canonical map
2071Equationhodge-theory-iii-5.3.3.1hodge-theory-iii-5.3.3.1.xml5.3.3.1hodge-theory-iii-5.3.3 \mathrm {R}\Gamma (S,K) \to \mathrm {R}\Gamma (S,\mathrm {R} a_*a^*K) \simeq \mathrm {R}\Gamma (X_\bullet ,a^*K) \tag{5.3.3.1}
is an isomorphism.
In particular, for F an abelian sheaf on S, we have a spectral sequence ()
2072Equationhodge-theory-iii-5.3.3.2hodge-theory-iii-5.3.3.2.xml5.3.3.2hodge-theory-iii-5.3.3 E_1^{pq} = \operatorname {H}^q(X_p,a_p^*F) \Rightarrow \operatorname {H}^{p+q}(S,F). \tag{5.3.3.2}
For a complex, we again have, in hypercohomology, a spectral sequence
2073Equationhodge-theory-iii-5.3.3.3hodge-theory-iii-5.3.3.3.xml5.3.3.3hodge-theory-iii-5.3.3 E_1^{pq} = \operatorname {\mathbb {H}}^q(X_p,a_p^*K) \Rightarrow \operatorname {\mathbb {H}}^{p+q}(S,K). \tag{5.3.3.3}
In both cases, the E_1^{\bullet q} (for fixed q) form a simplicial group, and
d_1 = \sum _i (-1)^i\delta _i \colon E_1^{p,q} \to E_1^{p+1,q}. 2108Definitionhodge-theory-iii-5.3.4hodge-theory-iii-5.3.4.xml5.3.4hodge-theory-iii-5.3
A continuous map a\colon X\to S is of cohomological descent if the augmentation morphism of \operatorname {cosq}(X\to S), namely
\big ((X/S)^{\Delta _n}\big )_{n\geqslant 0} \to S,
is of cohomological descent.
We say that a is of universal cohomological descent if, for every u\colon S'\to S, the continuous map a'\colon X\times _S S'\to S' is of cohomological descent.
2109hodge-theory-iii-5.3.5hodge-theory-iii-5.3.5.xml5.3.5hodge-theory-iii-5.3
The fundamental results, proven in [SD], are the following.
The continuous maps of universal cohomological descent form a Grothendieck topology on the category of topological spaces, which we call the universal cohomological descent topology.
A proper () surjective map is of universal cohomological descent.
A map a\colon X\to S that admits sections locally on S is of universal cohomological descent.
Let a\colon X_\bullet \to S be a k-truncated augmented simplicial space (with -1\leqslant k\leqslant \infty ).
For k\geqslant n\geqslant -1, let \varphi _n\colon \operatorname {cosq} X_\bullet \to \operatorname {cosq}\operatorname {sq}_n X_\bullet be the evident map.
We say that X_\bullet is a k-truncated hypercover of S, for the universal cohomological descent topology, if the maps
1909Equationhodge-theory-iii-5.3.5.1hodge-theory-iii-5.3.5.1.xml5.3.5.1hodge-theory-iii-5.3.5 (\varphi _n)_{n+1} \colon X_{n+1} \to (\operatorname {cosq}\operatorname {sq}_n X_\bullet )_{n+1} \tag{5.3.5.1}
(for -1\leqslant n\leqslant k-1) are of universal cohomological descent.
If X_\bullet is such a hypercover, then the simplicial space \operatorname {cosq}(X_\bullet ) augmented over S is of cohomological descent.
Let a be a morphism of simplicial topological spaces augmented over S.
\begin {CD} X_\bullet @>a>> Y_\bullet \\@VxVV @VVyV \\S @= S \end {CD}
We say that a is a hypercover for the universal cohomological descent topology if the evident maps X_n\to (\operatorname {cosq}_{n-1}^{Y_\bullet }\operatorname {sq}_{n-1}X_\bullet )_n are of universal cohomological descent.
If a is such a hypercover then, for every K\in \operatorname {Ob}\mathrm {D}^+(S),
a^*\colon \mathrm {R} y_*y^*K \xrightarrow {\sim } \mathrm {R} x_*x^*K.
2110hodge-theory-iii-5.3.6hodge-theory-iii-5.3.6.xml5.3.6hodge-theory-iii-5.3
For k=\infty , (IV) of implies that the a\colon X_\bullet \to S are of cohomological descent if the (\varphi _n)_{n+1} are of universal cohomological descent.
For n=-1,0, these maps are
(\varphi _n)_{n+1} = \begin {cases} X_0\xrightarrow {a}S &\text {if }n=-1 \\X_1\xrightarrow {(\delta _0,\delta _1)}X_0\times _S X_0 &\text {if }n=0. \end {cases}
For n=1, (\operatorname {cosq}\operatorname {sq}_1(X_\bullet ))_1 is the subspace of X_1\times _S X_1\times _S X_1 consisting of the triples (x,y,z) such that \delta _0x=\delta _0y, \delta _1x=\delta _0z, and \delta _1y=\delta _1z.
The map (\varphi _1)_2 is x\mapsto (\delta _0x,\delta _1x,\delta _2x).
For k=0, (IV) of is .
2111Examplehodge-theory-iii-5.3.7hodge-theory-iii-5.3.7.xml5.3.7hodge-theory-iii-5.3
Let \mathscr {U}=(U_i)_{i\in I} be an open cover, or a finite locally closed cover, of S.
Let X=\coprod _{i\in I}U_i.
Then a\colon X\to S is of cohomological descent.
The spectral sequence in for X_\bullet =\operatorname {cosq}(X\to S) is then exactly the Leray spectral sequence of the cover \mathscr {U}.
2112hodge-theory-iii-5.3.8hodge-theory-iii-5.3.8.xml5.3.8hodge-theory-iii-5.3
Let a\colon X_\bullet \to S be as in (IV) of .
We say that X_\bullet is a proper k-truncated hypercover of S if the arrows in are proper and surjective.
For k=\infty , we simply say "proper hypercover".
3015indexindex.xmlDeligne's "Hodge Theory I, II, and III"
This is an English translation of P. Deligne's three papers on Hodge theory:
"Théorie de Hodge I", Actes du Congrès intern. math. 1 (1970) pp. 425–430.
[PDF]
"Théorie de Hodge II", Pub. Math. de l'IHÉS 40 (1971) pp. 5–58.
[PDF]
"Théorie de Hodge III", Pub. Math. de l'IHÉS 44 (1974) pp. 5–77.
[PDF]TO-DO: replace fref by ref
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.
Hodge Theory I ✓
Hodge Theory II ✓
Introduction
Filtrations
Filtered objects
Opposite filtrations
The two filtrations lemma
Hypercohomology of filtered complexes
Hodge structures
Pure structures
Hodge theory
Mixed structures
Hodge theory of non-singular algebraic varieties
Logarithmic poles and residues
Mixed Hodge theory
Applications and supplements
The fixed set theorem
The semi-simplicity theorem
Supplement to [D1968]
Homomorphisms of abelian schemes
Hodge Theory III
Introduction ✓
Terminology and notation ✓
Cohomological descent ✓
Simplicial topological spaces
Cohomology of simplicial topological spaces
Cohomological descent
Examples of simplicial topological spaces ✓
Classifying spaces
Construction of hypercovers
Relative cohomology
Multi-simplicial spaces
Supplements to §1
Filtered derived category ✓
Supplements to the two-filtrations lemma
Hodge theory of algebraic spaces
Hodge complexes
Separated algebraic spaces
Hodge theory of simplicial schemes
Examples and applications
Cohomology of groups and of classifying spaces
Hodge theory of smooth hypersurfaces, following Griffiths
Construction of complexes of first-order differential operators
Hodge theory in level \leqslant 1
1-motives
1-motives and bi-extensions
Algebraic interpretation of the mixed \operatorname {H}^1: the case of curves
Translation of a theorem of Picard
Bibliography3016hodge-theory-iii-6hodge-theory-iii-6.xmlHodge Theory III › Examples of simplicial topological spaces6hodge-theory-iii2751hodge-theory-iii-6.1hodge-theory-iii-6.1.xmlClassifying spaces6.1hodge-theory-iii-62172hodge-theory-iii-6.1.1hodge-theory-iii-6.1.1.xml6.1.1hodge-theory-iii-6.1
Let u\colon X\to S be a continuous map.
For every sheaf F on S, the sheaf u^*F is endowed with a "descent data" with respect to u, i.e. we have an isomorphism between the two inverse images of u^*F on X\times _S X, and this isomorphism satisfies a cocycle condition.
If u admits a section locally on S, then this construction defines an equivalence between the category of sheaves on S and that of sheaves on X endowed with a descent data.
Take X to be a (left) principal homogeneous space for the group G on a space S.
Then the G-equivariant sheaves on X are exactly the sheaves endowed with a descent data: every equivariant sheaf on X is, in a unique way (as an equivariant sheaf), the inverse image of a sheaf on S=X/G.
2173hodge-theory-iii-6.1.2hodge-theory-iii-6.1.2.xml6.1.2hodge-theory-iii-6.1
If a topological group G acts on a space X, then G acts on G^{\Delta _n}\times X by
g\cdot (g_0,\ldots ,g_n,x) = (g_0g^{-1},\ldots ,g_ng^{-1},gx).
We denote by [X/G]_\bullet the simplicial space
2138Equationhodge-theory-iii-6.1.2.1hodge-theory-iii-6.1.2.1.xml6.1.2.1hodge-theory-iii-6.1.2 [X/G]_\bullet = \big ((G^{\Delta _n}\times X)/G\big )_{n\geqslant 0}. \tag{6.1.2.1}
If X is a principal homogeneous space for the group G on S=X/G, then the map
\begin {aligned} G^{\Delta _n}\times X &\to X^{\Delta _n} \\(g_0,\ldots ,g_n,x) &\mapsto (g_0x,\ldots ,g_nx) \end {aligned}
identifies [X/G]_n with the iterated fibre product (X/S)^{\Delta _n}:
\operatorname {cosq}(X\to S) = ([X/G]_\bullet \to S).
In particular, we have (by (III) of )
2141Equationhodge-theory-iii-6.1.2.2hodge-theory-iii-6.1.2.2.xml6.1.2.2hodge-theory-iii-6.1.2 \operatorname {H}^\bullet ([X/G]_\bullet ) \simeq \operatorname {H}^\bullet (X/G) \tag{6.1.2.2}
(for a principal homogeneous space).
For all n, G^{\Delta _n}\times X is a principal homogeneous space for the group G on [X/G]_n.
For every equivariant sheaf F on X, \operatorname {pr}_2^*F is an equivariant sheaf on G^{\Delta _n}\times X;
by , the latter is the inverse image of F^n on [X/G]_n.
Every equivariant sheaf F on X thus defines a sheaf on [X/G]_\bullet .
It is easy to show that we thus obtain an equivalence between the category of equivariant sheaves on X and the category of sheaves F^\bullet on [X/G]_\bullet that satisfy
2143Propertyhodge-theory-iii-6.1.starhodge-theory-iii-6.1.star.xml*hodge-theory-iii-6.1.2(*) For every f\colon \Delta _n\to \Delta _m, the structure morphism f^*F^n\to F^m is an isomorphism.
Construction (b) above is natural in (G,X,F).
We set
2145Equationhodge-theory-iii-6.1.2.3hodge-theory-iii-6.1.2.3.xml6.1.2.3hodge-theory-iii-6.1.2 \operatorname {H}^\bullet (X,G;F) = \operatorname {H}^\bullet ([X/G]_\bullet ,F^\bullet ) \tag{6.1.2.3}
(mixed cohomology of X,G with coefficients in F).
Under the hypotheses of (a), if F is the inverse image of F^{-1} on S=X/G, then
2146Equationhodge-theory-iii-6.1.2.4hodge-theory-iii-6.1.2.4.xml6.1.2.4hodge-theory-iii-6.1.2 \operatorname {H}^\bullet (X,G;F) = \operatorname {H}^\bullet (X/G,F^{-1}) \tag{6.1.2.4}
(for a principal homogeneous space).
This generalises (which is the case F=\underline {\mathbb {Z}}).
2175hodge-theory-iii-6.1.3hodge-theory-iii-6.1.3.xml6.1.3hodge-theory-iii-6.1
Let P^t be the topological space consisting of a single point.
We define the simplicial classifying space of G, denoted B_{\bullet G}, to be the simplicial space
B_{\bullet G} = [P^t/G]_\bullet .
Let P be a principal homogeneous space for the group G on S.
The evident morphism
\operatorname {cosq}(P\to S) = [P/G]_\bullet \to [P^t/G]_\bullet = B_{\bullet G}
defines a composite morphism
2174Equationhodge-theory-iii-6.1.3.1hodge-theory-iii-6.1.3.1.xml6.1.3.1hodge-theory-iii-6.1.3 [P] \colon \operatorname {H}^\bullet (B_{\bullet G}) \to \operatorname {H}^\bullet ([P/G]_\bullet ) \xleftarrow [(6.1.2.2)]{\sim } \operatorname {H}^\bullet (S). \tag{6.1.3.1}
We will see below that, in good cases, \operatorname {H}^\bullet (B_{\bullet G})=\operatorname {H}^\bullet (B_G), and that the image of [P] consists of the characteristic classes of P.
2178hodge-theory-iii-6.1.4hodge-theory-iii-6.1.4.xml6.1.4hodge-theory-iii-6.1
Let G be a Lie group, B_G a classifying space for G, and a\colon U_G\to B_G the universal principal homogeneous G-space.
Let X be a G-space;
then X\times U_G is a principal homogeneous G-space over X\times U_G/G such that, for every equivariant sheaf F on X, \operatorname {pr}_1^*F is the inverse image of a sheaf F^G on X\times U_G/G.
Since U_G is contractible, and by , we have
2176Equationhodge-theory-iii-6.1.4.1hodge-theory-iii-6.1.4.1.xml6.1.4.1hodge-theory-iii-6.1.4 \operatorname {H}^\bullet (X,G;F) \xrightarrow {\sim } \operatorname {H}^\bullet (X\times U_G,G;\operatorname {pr}_1^*F) \xleftarrow {\sim } \operatorname {H}^\bullet (X\times U_G/G,F^G). \tag{6.1.4.1}
In particular, for X=P^t,
2177Equationhodge-theory-iii-6.1.4.2hodge-theory-iii-6.1.4.2.xml6.1.4.2hodge-theory-iii-6.1.4 \operatorname {H}^\bullet (B_{\bullet G}) = \operatorname {H}^\bullet (B_G). \tag{6.1.4.2}
We can see that the isomorphism in is a particular case of where S=B_G and P=U_G.
2179hodge-theory-iii-6.1.5hodge-theory-iii-6.1.5.xml6.1.5hodge-theory-iii-6.1
The spectral sequence in for B_{\bullet G}
E_1^{pq} = \operatorname {H}^q(G^{\Delta _p}/G) \Rightarrow \operatorname {H}^{p+q}(B_{\bullet G}) =\operatorname {H}^{p+q}(B_G)
is essentially the Eilenberg–Moore spectral sequence.
We briefly recall how it allows us to relate the rational cohomologies of G and of B_G, for connected G.
The algebra \operatorname {H}^\bullet (G,\mathbb {Q}) is a connected graded Hopf algebra of finite dimension over \mathbb {Q}.
If P^\bullet (G) is the graded module of its primitive elements, then we have
H^\bullet (G,\mathbb {Q}) = \bigwedge P^\bullet (G)
and the generators of P^\bullet (G) are of odd degree.
The simplicial algebra (E_1^{p\bullet })_{p\geqslant 0} is
E_1^{p\bullet } = \bigwedge \big (P^\bullet (G)^{\Delta _p}/P^\bullet (G)\big )
which is the exterior algebra of the suspension of the constant cosimplicial module P^\bullet (G);
we thus have (by Quillen [Q1968]) that E_2^{p\bullet }=\operatorname {Sym}^p(P^\bullet (G)).
The pages E_2^{pq} are only zero for p+q even;
we thus have that E_2^{pq}=E_\infty ^{pq}, and, for a suitable filtration, we canonically have that
\operatorname {Gr}\operatorname {H}^\bullet (B_G,\mathbb {Q}) \simeq \operatorname {Sym}^\bullet (P^\bullet (G)[-1])
and non-canonically that
\operatorname {H}^\bullet (B_G,\mathbb {Q}) \simeq \operatorname {Sym}^\bullet (P^\bullet (G)[-1]).
2180hodge-theory-iii-6.1.6hodge-theory-iii-6.1.6.xml6.1.6hodge-theory-iii-6.1
Let G be a (complex) linear algebraic group.
If T is a maximal torus of G, with Weyl group W, then
253Equationhodge-theory-iii-6.1.6.1hodge-theory-iii-6.1.6.1.xml6.1.6.1hodge-theory-iii-6.1.6 \operatorname {H}^\bullet (B_G,\mathbb {Q}) \xrightarrow {\sim } \operatorname {H}^\bullet (B_T,\mathbb {Q})^W. \tag{6.1.6.1}
If T is a torus with character group X(T), then
255Equationhodge-theory-iii-6.1.6.2hodge-theory-iii-6.1.6.2.xml6.1.6.2hodge-theory-iii-6.1.6 \operatorname {H}^\bullet (T,\mathbb {Z}) \simeq \bigwedge ^\bullet X(T) \tag{6.1.6.2}
(an isomorphism of graded Hopf algebras).
We will only use (a) in the following weaker form:
(The splitting principle).
The map \operatorname {H}^\bullet (B_G,\mathbb {Q})\to \operatorname {H}^\bullet (B_T,\mathbb {Q}) is injective.
For completion, we recall a proof of (a').
If B is a Borel subgroup of G, then the bundle U_G/B on B_G is a fibre in flag spaces.
By [2.1 and 2.6.3, D1968], which is better explained in [Proposition 3.1, G1970], or by [B1956], the Leray spectral sequence of U_G/B\to B_G degenerates to rational cohomology.
We thus have that \operatorname {H}^\bullet (B_G,\mathbb {Q})\hookrightarrow \operatorname {H}^\bullet (U_G/B,\mathbb {Q}).
We conclude by noting that U_G/B\sim B_B\sim B_T.
2752hodge-theory-iii-6.2hodge-theory-iii-6.2.xmlConstruction of hypercovers6.2hodge-theory-iii-62234hodge-theory-iii-6.2.1hodge-theory-iii-6.2.1.xml6.2.1hodge-theory-iii-6.2
Let X_\bullet be a simplicial set.
Denote by D(\Delta _n,\Delta _M) the set of increasing surjective maps from \Delta _n to \Delta _m (degeneracy operators), and set
2232Equationhodge-theory-iii-6.2.1.1hodge-theory-iii-6.2.1.1.xml6.2.1.1hodge-theory-iii-6.2.1 N(X_n) = X_n\setminus \bigcup _{{s\in D(\Delta _n,\Delta _{n-1})}}s(X_{n-1}). \tag{6.2.1.1}
Recall that, for all n, the map
2233Equationhodge-theory-iii-6.2.1.2hodge-theory-iii-6.2.1.2.xml6.2.1.2hodge-theory-iii-6.2.1 \coprod s \colon \coprod _{\mathclap {\substack {m\leqslant n,\\s\in D(\Delta _n,\Delta _m)}}} N(X_m) \to X_n \tag{6.2.1.2}
is bijective.
2235Definitionhodge-theory-iii-6.2.2hodge-theory-iii-6.2.2.xml6.2.2hodge-theory-iii-6.2
We say that a simplicial topological space is s-split if the maps in are homeomorphisms.
Let X_\bullet be a k-truncated simplicial set.
For n\leqslant k, we again define N(X_n) by , and then is a bijection.
We say that a k-truncated simplicial topological space is s-split if is a homeomorphism for n\leqslant k.
2237hodge-theory-iii-6.2.3hodge-theory-iii-6.2.3.xml6.2.3hodge-theory-iii-6.2
For an s-split (n+1)-truncated topological simplicial space X augmented over S, let \alpha (X) be the triple consisting of \operatorname {sq}_n(X), NX_{n+1}, and the evident map from NX_{n+1} to (\operatorname {cosq}\operatorname {sq}_n X)_{n+1}.
This triple \alpha (X)=(Y,N,\beta ) satisfies the following:
2236Propertyhodge-theory-iii-6.2.starhodge-theory-iii-6.2.star.xml*hodge-theory-iii-6.2.3(*) Y is an s-split n-truncated simplicial topological space augmented over S, and \beta is a continuous map from N to (\operatorname {cosq} Y)_{n+1}.
2246Propositionhodge-theory-iii-6.2.4hodge-theory-iii-6.2.4.xml6.2.4hodge-theory-iii-6.2
Let (Y,N,\beta ) satisfy above.
Up to unique isomorphism, there exists exactly one s-split (n+1)-truncated topological space X augmented over S such that \alpha (X)\simeq (Y,N,\beta ).
It is equivalent to give either f\colon X\to Z or:
a morphism f'\colon Y\to \operatorname {sq}_n(Z); and
a morphism f''\colon N\to Z_{n+1}, such that the diagram
\begin {CD} N @>\beta >> (\operatorname {cosq} Y)_{n+1} \\@V{f''}VV @VV{f'}V \\Z_{n+1} @>>> (\operatorname {cosq}\operatorname {sq}_n Z)_{n+1} \end {CD}
commutes.
2244Proof#177unstable-177.xmlhodge-theory-iii-6.2.4
[5.1.3, SD].
This proposition also applies to simplicial objects in other categories \mathscr {C} apart from that of topological spaces; it suffices that \mathscr {C} satisfy the following:
2245hodge-theory-iii-6.2.4.1hodge-theory-iii-6.2.4.1.xml6.2.4.1hodge-theory-iii-6.2.4
Finite projective limits exist in \mathbb {C}.
Finite projective sums exist in \mathbb {C}, and they are disjoint and universal.
2247hodge-theory-iii-6.2.5hodge-theory-iii-6.2.5.xml6.2.5hodge-theory-iii-6.2 allows us to construct, by induction, proper hypercovers of S.
We take f_0\colon X_0\to S, proper and surjective.
Then \{X_0\} is a 0-truncated proper hypercover of S (), and it is s-split.
We take f_1\colon N_1\to \operatorname {cosq}(\{X_0\})_1, i.e. f_1\colon N_1\to X_0\times _S X_0.
Applying , we associate to f_1 the s-split 1-truncated augmented simplicial topological space
{}_1X_\bullet = \left ( N_1{\textstyle \coprod } X_0 \xleftarrow [\to ]{\to } X_0 \to S \right ).
Suppose f_1 to be chosen such that
f'_1 \colon N_1{\textstyle \coprod } X_0\to X_0\times _S X_0
is proper and surjective (for example, if f_1 is proper and surjective).
Then {}_1X_\bullet is an s-split 1-truncated proper hypercover of S.
Assume that we have already constructed an s-split k-truncated proper hypercover {}_kX_\bullet \to S.
We take f_{k+1}\colon N_{k+1}\to (\operatorname {cosq}({}_kX_\bullet ))_{k+1} and, applying , we associate to f_{k+1} an s-split (k+1)-truncated augmented semi-simplicial space {}_{k+1}X_\bullet .
Suppose that f_{k+1} is such that
f'_{k+1} \colon {}_{k+1}X_{k+1} \to \operatorname {cosq}({}_kX_\bullet )_{k+1}
is proper and surjective (for example, if f_{k+1} is proper and surjective).
Then {}_{k+1}X_\bullet is an s-split (k+1)-truncated proper hypercover of S.
The {}_kX_\bullet thus constructed are the successive skeletons of an s-split proper hypercover of S.
2248hodge-theory-iii-6.2.6hodge-theory-iii-6.2.6.xml6.2.6hodge-theory-iii-6.2
We say that a simplicial scheme X_\bullet over \mathbb {C} is smooth if the X_n are smooth;
it is said to be proper if the X_n are compact.
A normal crossing divisor D_\bullet of X_\bullet , assumed to be smooth, is a family D_n\subset X_n of normal crossing divisors () such that the U_n=X_n\setminus D_n form a simplicial subscheme U_\bullet of X_\bullet .
This definition is justified by the following lemma.
2249Lemmahodge-theory-iii-6.2.7hodge-theory-iii-6.2.7.xml6.2.7hodge-theory-iii-6.2
If D_\bullet is a normal crossing divisor of X_\bullet , then the logarithmic de Rham complexes (\Omega _{X_n}^\bullet (\log D_n))_{n\geqslant 0}, endowed with the weight filtration (), form a filtered complex on X_\bullet .
1378Proof#178unstable-178.xmlhodge-theory-iii-6.2.7
This follows from (ii) of .
We denote the complex (\Omega _{X_n}^\bullet (\log D_n))_{n\geqslant 0} by \Omega _{X_\bullet }^\bullet (\log D_\bullet ).
2254hodge-theory-iii-6.2.8hodge-theory-iii-6.2.8.xml6.2.8hodge-theory-iii-6.2
Using , we can show that, for every separated scheme S over \mathbb {C}, there exists:
a simplicial scheme X_\bullet over \mathbb {C}, smooth and proper, that we can take to be s-split;
a normal crossing divisor D_\bullet of X_\bullet ; we set U_\bullet =X_\bullet \setminus D_\bullet ;
an augmentation a\colon U_\bullet \to S that realises U_\bullet ^\mathrm {an} as a proper hypercover of S^\mathrm {an}.
Furthermore, any two such systems are covered by a third, and a morphism u\colon S\to T can be covered by a morphism
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
X_\bullet
\ar [r]
& Y_\bullet
\\U_\bullet
\ar [r]
\ar [u,hook]
\ar [d,swap,"a"]
& V_\bullet
\ar [u,hook]
\ar [d,"b"]
\\S \ar [r,swap,"u"]
& T
\end {tikzcd}
of such systems (see [SD]).
2753hodge-theory-iii-6.3hodge-theory-iii-6.3.xmlRelative cohomology6.3hodge-theory-iii-62302hodge-theory-iii-6.3.1hodge-theory-iii-6.3.1.xml6.3.1hodge-theory-iii-6.3
The mapping cone construction for morphisms of simplicial sets works in the same way for simplicial objects in any category \mathscr {C} that has a final object e and finite sums.
For u\colon Y_\bullet \to X_\bullet , the cone C(u) satisfies
C(u)_n = X_n{\,\textstyle \coprod \,}\coprod _{i<n} Y_i{\,\textstyle \coprod \,}e.
We take \mathscr {C} to be:
the category of topological spaces and continuous maps, with final object e=P^t;
the category of pairs (X,F) where X is a topological space and F is an abelian sheaf on X, with an arrow (u,f)\colon (Y,F)\to (X,G) consisting of a continuous map u\colon Y\to X and a u-morphism () f\colon G\to F, with final object e=(P^t,0).
2305hodge-theory-iii-6.3.2hodge-theory-iii-6.3.2.xml6.3.2hodge-theory-iii-6.3
Let u\colon Y_\bullet \to X_\bullet be a morphism of topological simplicial spaces, with cone C(u).
Let F be an abelian sheaf on X_\bullet and G an abelian sheaf on Y_\bullet , and let f\colon G\to F be a u-morphism.
The cone C(f) of f is an abelian sheaf on C(u), and we set
2303Equationhodge-theory-iii-6.3.2.1hodge-theory-iii-6.3.2.1.xml6.3.2.1hodge-theory-iii-6.3.2 \operatorname {H}^n(C(u),C(f)) = \operatorname {H}^n(X_\bullet \operatorname {mod} Y_\bullet , F\operatorname {mod} G). \tag{6.3.2.1}
These are the relative cohomology groups.
We can easily show that they fit into a long exact sequence
2304Equationhodge-theory-iii-6.3.2.2hodge-theory-iii-6.3.2.2.xml6.3.2.2hodge-theory-iii-6.3.2 \ldots \to \operatorname {H}^i(X_\bullet \operatorname {mod} Y_\bullet , F\operatorname {mod} G) \to \operatorname {H}^i(X_\bullet ,F) \to \operatorname {H}^i(Y_\bullet ,G) \to \ldots . \tag{6.3.2.2} 2307hodge-theory-iii-6.3.3hodge-theory-iii-6.3.3.xml6.3.3hodge-theory-iii-6.3
More generally, let L and K be bounded-below complexes of abelian sheaves on Y_\bullet and X_\bullet (respectively), and let f\colon K\to L be a u-morphism.
We thus obtain a complex C(f) on C(u).
We again define the hypercohomology
\operatorname {\mathbb {H}}^n(C(u),C(f)) = \operatorname {\mathbb {H}}^n(X_\bullet \operatorname {mod} Y_\bullet , K\operatorname {mod} L).
These groups appear in an exact sequence analogous to that of , coming, in the suitable derived category, from a distinguished triangle
2306Equationhodge-theory-iii-6.3.3.1hodge-theory-iii-6.3.3.1.xml6.3.3.1hodge-theory-iii-6.3.3
\usepackage {amsmath}
\usepackage {amssymb}
\usepackage {tikz-cd}
\begin {tikzcd}
& \mathrm {R}\Gamma (Y_\bullet ,K)
\ar [dl,swap,"\partial "]
\\\mathrm {R}\Gamma (C(u),C(f))
\ar [rr]
&& \mathrm {R}\Gamma (X_\bullet ,L)
\ar [ul]
& (6.3.3.1)
\end {tikzcd}
2308hodge-theory-iii-6.3.4hodge-theory-iii-6.3.4.xml6.3.4hodge-theory-iii-6.3
The construction presented above is not the only one possible.
It has the inconvenience that, even if we start with true topological spaces X and Y (i.e. constant simplicial spaces), we are led to consider non-constant simplicial spaces.
2754hodge-theory-iii-6.4hodge-theory-iii-6.4.xmlMultisimplicial spaces6.4hodge-theory-iii-62325hodge-theory-iii-6.4.1hodge-theory-iii-6.4.1.xml6.4.1hodge-theory-iii-6.4
Let r\geqslant 0 be an integer.
An r-simplicial object Z_\bullet in a category \mathscr {C} is a contravariant functor from the r-fold product (\Delta )^r to \mathscr {C}.
The diagonal simplicial object \delta Z_\bullet is the composite functor (\Delta )\to (\Delta )^r\to \mathscr {C}.
2326hodge-theory-iii-6.4.2hodge-theory-iii-6.4.2.xml6.4.2hodge-theory-iii-6.4
As in , we define the topos of sheaves on an r-simplicial topological space.
Let \Gamma ^\bullet be the functor
F \mapsto (\Gamma (X_{n_1\ldots n_r},F^{n_1\ldots n_r}))
from sheaves on X_\bullet to r-cosimplicial sets.
For small r, we often prefer to write \Gamma ^{\bullet \ldots \bullet } (with r-many copies of \bullet ).
We have a co-augmentation \Gamma (X_\bullet ,F^\bullet )\to \Gamma ^\bullet (X_\bullet ,F^\bullet ).
The functors of cohomology with values in an abelian sheaf F are the derived functors of the "global sections" functor \Gamma .
They can be calculated by a procedure parallel to that of , i.e.
2014Equationhodge-theory-iii-6.4.2.1hodge-theory-iii-6.4.2.1.xml6.4.2.1hodge-theory-iii-6.4.2 \mathrm {R}\Gamma =\mathrm {s}\mathrm {R}\Gamma ^\bullet \colon \mathrm {D}^+(X_\bullet ) \to \mathrm {D}^+((\mathsf {Ab})). \tag{6.4.2.1}
A sheaf F on an r-simplicial topological space X_\bullet induces a sheaf \delta F on the diagonal simplicial space \delta X_\bullet .
It follows from the Cartier–Eilenberg–Zilber theorem that
2015Equationhodge-theory-iii-6.4.2.2hodge-theory-iii-6.4.2.2.xml6.4.2.2hodge-theory-iii-6.4.2 \mathrm {R}\Gamma (X_\bullet ,K) \xrightarrow {\sim } \mathrm {R}\Gamma (\delta X_\bullet ,\delta K). \tag{6.4.2.2} 2327hodge-theory-iii-6.4.3hodge-theory-iii-6.4.3.xml6.4.3hodge-theory-iii-6.4
We restrict to the case r=2.
A bisimplicial object 1-augmented over a simplicial object S_\bullet is a contravariant functor from (\Delta ^+)\times (\Delta ) to \mathscr {C} such that S_\bullet is the composite functor
(\Delta ) \xrightarrow {(\Delta _{-1},\bullet )} (\Delta ^+)\times (\Delta ) \to \mathscr {C}.
To denote a bisimplicial object 1-augmented over S_\bullet , with underlying bisimplicial object X_{\bullet \bullet }, we will use notation of the form
a\colon X_{\bullet \bullet } \to S_\bullet .
For n\geqslant 0, a\colon (X_{\bullet n})\to S_n is a simplicial object augmented over S_n.
If F is a sheaf on X_{\bullet \bullet }, then the (a_*(F|X_{\bullet n})\text { on }S_n)_{n\geqslant 0} form a sheaf on S_\bullet ;
we thus define a topos morphism
a \colon X_{\bullet \bullet }^\sim \to S_\bullet ^\sim .
We explain in [SD] that \mathrm {R} a_* can be calculated "component by component", i.e.
\mathrm {R} a_*K|S_n = \mathrm {R} a_*(K|X_{\bullet n}).
It thus follows that if, for each n, the morphism a\colon (X_{\bullet n})\to S_n is of cohomological descent, then a\colon X_{\bullet \bullet }\to S_\bullet is of cohomological descent: for every complex K\in \mathrm {D}^+(S_\bullet ) of abelian sheaves, we have
K \xrightarrow {\approx } \mathrm {R} a_*a^*K. 2328hodge-theory-iii-6.4.4hodge-theory-iii-6.4.4.xml6.4.4hodge-theory-iii-6.4
In [SD], we show that, for every separated simplicial scheme S_\bullet , there exists a bisimplicial scheme X_{\bullet \bullet } that is 1-augmented over S_\bullet , along with i\colon X_{\bullet \bullet }\hookrightarrow \bar {X}_{\bullet \bullet } such that:
The \bar {X}_{nm} are smooth and projective;
X_{nm} is the complement of a normal crossing divisor D_{nm} in \bar {X}_{nm}, and we can suppose the D_{nm} to be a union of smooth divisors.
For n\geqslant 0, X_{\bullet n} is a proper hypercover of S_n, and we can take it to be s-split.
The construction proceeds as in , but the induction is more complicated.
The claims of uniqueness () remain true, mutatis mutandis.
3017hodge-theory-iii-introductionhodge-theory-iii-introduction.xmlHodge Theory III › Introductionhodge-theory-iii
This article follows Hodge Theory I and Hodge Theory II;
the numbering of the sections here follows that of (which contains to ).
In , we gave an introduction to the yoga that underlies and the present article .
However, and are logically independent of ;
they also do not contain all the results that were stated in .
In , we introduced the Hodge theory of non-singular (not necessarily complete) algebraic varieties.
Here we treat the case of arbitrary singularities.
From [hodge-theory-iii-7 (?)] onwards, we will make essential use of results from to .
In the case of a complete singular algebraic variety X, the fundamental idea is to use the resolution of singularities to replace X by a simplicial system of smooth projective schemes
\usepackage {amsmath}
\usepackage {amssymb}
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\begin {tikzcd}
X_\bullet = \Big (\ldots \quad
X_2
\ar [r]
\ar [r,shift left=4]
\ar [r,shift right=4]
& X_1
\ar [r,shift left=2]
\ar [r,shift right=2]
\ar [l,shift right=2]
\ar [l,shift left=2]
& X_0\Big )
\ar [l]
\end {tikzcd}
(or, as we will say, by a smooth projective simplicial scheme).
The results of [hodge-theory-iii-5 (?)] and [hodge-theory-iii-6 (?)] allow us to find such simplicial schemes X_\bullet that have, in a suitable sense, the same cohomology as X.
We can then "express", via a spectral sequence, the cohomology of X in terms of the cohomology of the X_n.
Following the general principles of , we can then, on each X_n, use classical Hodge theory, and obtain an induced mixed Hodge structure on the cohomology of X.
In the case of an arbitrary algebraic variety X, we "replace" X by a smooth simplicial scheme X_\bullet with compactification given by a smooth projective simplicial scheme \bar {X}_\bullet .
We can further arrange things so that \bar {X}_n\setminus X_n is a normal crossing divisor, given by a union of smooth divisors D_{n,i}.
We thus "express", via a spectral sequence, the cohomology of X in terms of the cohomology of the p-fold intersections of the D_{n,i} (for n,p\geqslant 0), and we obtain an induced mixed Hodge structure on H^\bullet (X).
[hodge-theory-iii-5 (?)] and [hodge-theory-iii-6 (?)] constitute a summary (without proofs) of the theory of "cohomological descent".
This theory, in the more general framework of a topos, is laid out in [SD].
In [hodge-theory-iii-7.1 (?)], we restate certain results obtained previously, but in the language of filtered derived categories;
[hodge-theory-iii-7.2 (?)] contains a simpler proof of the two-filtrations lemma than that of .
The unappealing [hodge-theory-iii-8.1 (?)] is the heart of this work.
The essential result is [hodge-theory-iii-8.1.15 (?)].
In its proof, to be able to apply the two-filtrations lemma, we use the fact that every morphism of mixed Hodge structures is strictly compatible with the filtrations W and W ().
In [hodge-theory-iii-8.2 (?)], we define the mixed Hodge structure of \operatorname {H}^n(X,\mathbb {Z}) (for X a separated scheme).
We show that the Hodge numbers h^{pq} of \operatorname {H}^n(X,\mathbb {Z}) can only be non-zero for (p,q)\in [0,n]\times [0,n].
For n>\dim X, we have a more precise result ([hodge-theory-iii-8.2.4 (?)]).
If X is smooth (resp. complete), they can only be non-zero if, further, p+q\geqslant n (resp. p+q\leqslant n).
If X is a complete subscheme of Z, with complement U, with Z complete and smooth of dimension n, then we can, by N. Katz, interpret this "duality" between the complete case and the smooth case as coming from the "Alexander duality"
\to \ldots \to \operatorname {H}^i(Z,\mathbb {Q}) \to \operatorname {H}^i(X,\mathbb {Q}) \to (\operatorname {H}^{2n-i-1}(U,\mathbb {Q}))^* \to \operatorname {H}^{i+1}(Z,\mathbb {Q}) \to \ldots .
From the functorial properties of the theory thus constructed, we obtain the useful corollaries [hodge-theory-iii-8.2.5 (?)] to [hodge-theory-iii-8.2.8 (?)], which we state in terms independent of any Hodge theory.
In [hodge-theory-iii-8.3 (?)], we endow the cohomology of any simplicial scheme X_\bullet with a mixed Hodge structure.
This generality is not illusory.
We can interpret relative cohomology spaces as being the cohomology of suitable simplicial schemes.
Let B_G be the classifying space of the underlying Lie group of an algebraic group G.
We can interpret the cohomology of B_G as being the cohomology of a suitable simplicial scheme.
In [hodge-theory-iii-9.1 (?)], after having calculated the mixed Hodge structure on the cohomology of B_G (for G linear), we thus obtain that of G.
As a corollary, we find that, if a linear group G acts on a non-singular complete variety X, then the map
\operatorname {H}^\bullet (X,\mathbb {Q}) \to \operatorname {H}^\bullet (X,\mathbb {Q})\otimes \operatorname {H}^\bullet (G,\mathbb {Q})
factors through \operatorname {H}^\bullet (X,\mathbb {Q})\otimes \operatorname {H}^0(G,\mathbb {Q})\subset \operatorname {H}^\bullet (X,\mathbb {Q})\otimes \operatorname {H}^\bullet (G,\mathbb {Q}).
In [hodge-theory-iii-10 (?)], we interpret in terms of abelian schemes the mixed Hodge structures of pure degree \{(-1,-1),(-1,0),(0,-1),(0,0)\} (see [hodge-theory-iii-0.5 (?)]), and we consider in detail the \operatorname {H}^1 of curves.
The theory developed up until here is an absolute theory (we do not study the functors \mathrm {R} f_*), and deals only with constant coefficients.
I conjecture that, if \mathscr {H} is a variation of polarisable Hodge structures (in the sense of Griffiths [G1970a]) on a scheme X, then the cohomology of X with coefficients in the local system \mathscr {H} is endowed with a natural mixed Hodge structure.
I can only prove this when X is complete.
3018hodge-theory-iii-7hodge-theory-iii-7.xmlHodge Theory III › Supplements to §17hodge-theory-iii2773hodge-theory-iii-7.1hodge-theory-iii-7.1.xmlFiltered derived category7.1hodge-theory-iii-7
This section completes .
2350hodge-theory-iii-7.1.1hodge-theory-iii-7.1.1.xml7.1.1hodge-theory-iii-7.1
Let \mathscr {A} be an abelian category.
We set:
\mathrm {F}\mathscr {A} (resp. \mathrm {F}_2\mathscr {A})
= the category of filtered (resp. bifiltered) objects with finite filtration(s) of \mathscr {A}.
\mathrm {K}^+\mathrm {F}\mathscr {A} (resp. \mathrm {K}^+\mathrm {F}_2\mathscr {A})
= the category of filtered (resp. bifiltered) bounded-below complexes of objects of \mathscr {A}, up to homotopy that respects the filtration(s).
\mathrm {D}^+\mathrm {F}\mathscr {A} (resp. \mathrm {D}^+\mathrm {F}_2\mathscr {A})
= the triangulated category induced from \mathrm {K}^+\mathrm {F}\mathscr {A} (resp. from \mathrm {K}^+\mathrm {F}_2\mathscr {A}) by inverting the filtered (resp. bifiltered) quasi-isomorphisms ();
this is the derived filtered category (resp. derived bifiltered category).
2351hodge-theory-iii-7.1.2hodge-theory-iii-7.1.2.xml7.1.2hodge-theory-iii-7.1
A filtered quasi-isomorphism u\colon (K,F)\to (K',F') induces an isomorphism of spectral sequences u\colon E_\bullet ^{\bullet \bullet }(K,F)\to E_\bullet ^{\bullet \bullet }(K',F').
An object K of \mathrm {D}^+\mathrm {F}\mathscr {A} thus defines a spectral sequence E_\bullet ^{\bullet \bullet }(K).
Similarly, an object L of \mathrm {D}^+\mathrm {F}_2\mathscr {A} defines an accumulation of spectral sequences of the type considered in .
2352hodge-theory-iii-7.1.3hodge-theory-iii-7.1.3.xml7.1.3hodge-theory-iii-7.1
Let T be a left exact functor from \mathscr {A} to an abelian category \mathscr {B}.
Suppose that every object of \mathscr {A} injects into an injective object.
The functor T can then be "derived" to give the functors
1063Equationhodge-theory-iii-7.1.3.1hodge-theory-iii-7.1.3.1.xml7.1.3.1hodge-theory-iii-7.1.3 \mathrm {R}\colon \mathrm {D}^+(\mathscr {A}) \to \mathrm {D}^+(\mathscr {B}) \tag{7.1.3.1}
1064Equationhodge-theory-iii-7.1.3.2hodge-theory-iii-7.1.3.2.xml7.1.3.2hodge-theory-iii-7.1.3 \mathrm {R}\colon \mathrm {D}^+\mathrm {F}(\mathscr {A}) \to \mathrm {D}^+\mathrm {F}(\mathscr {B}) \tag{7.1.3.2}
1065Equationhodge-theory-iii-7.1.3.3hodge-theory-iii-7.1.3.3.xml7.1.3.3hodge-theory-iii-7.1.3 \mathrm {R}\colon \mathrm {D}^+\mathrm {F}_2(\mathscr {A}) \to \mathrm {D}^+\mathrm {F}_2(\mathscr {B}). \tag{7.1.3.3}
They can be calculated as follows: if K' is a T-acyclic resolution (resp. a filtered resolution, resp. a bifiltered resolution) of K ( and ), then \mathrm {R} T(K)=T(K').
The hypercohomology spectral sequence (for T) of K\in \operatorname {Ob}\mathrm {D}^+\mathrm {F}(\mathscr {A}) is the spectral sequence of \mathrm {R} TK\in \operatorname {Ob}\mathrm {D}^+\mathrm {F}(\mathscr {B}) (cf. ).
2353hodge-theory-iii-7.1.4hodge-theory-iii-7.1.4.xml7.1.4hodge-theory-iii-7.1
We will need more precise results for the functors \mathrm {R} a_*, where a\colon X_\bullet \to S is an augmentation of a simplicial topological space.
The case S=P^t, where \mathrm {R} a_*=\mathrm {R}\Gamma will suffice.
We reuse the notation of , and recall :
\mathrm {R} a_* = \mathrm {s}\mathrm {R} {a_\bullet }_*.
For every complex K\in \operatorname {Ob}\mathrm {C}^+(S_\bullet ), the simple complex \mathrm {s} K is endowed with a natural filtration L ().
A quasi-isomorphism u\colon K'\xrightarrow {\sim } K'' induces a filtered quasi-isomorphism u\colon (\mathrm {s} K',L)\xrightarrow {\sim }(\mathrm {s} K'',L).
Then \mathrm {s} factors as
\mathrm {s}\colon \mathrm {D}^+(S_\bullet ) \to \mathrm {D}^+\mathrm {F}(S)
and \mathrm {R} a_* factors as
\mathrm {R} a_*\colon \mathrm {D}^+(X_\bullet ) \to \mathrm {D}^+\mathrm {F}(S).
The spectral sequence of the filtered complex (\mathrm {R} a_* K,L)\in \mathrm {D}^+\mathrm {F}(S) is exactly .
2356hodge-theory-iii-7.1.5hodge-theory-iii-7.1.5.xml7.1.5hodge-theory-iii-7.1
If K is filtered (resp. bifiltered), then \mathrm {R}{a_\bullet }_*K is filtered (resp. bifiltered): we have
2354Equationhodge-theory-iii-7.1.5.1hodge-theory-iii-7.1.5.1.xml7.1.5.1hodge-theory-iii-7.1.5 \mathrm {R}{a_\bullet }_* \colon \mathrm {D}^+\mathrm {F}(X_\bullet ) \to \mathrm {D}^+\mathrm {F}(S_\bullet ) \tag{7.1.5.1}
2355Equationhodge-theory-iii-7.1.5.2hodge-theory-iii-7.1.5.2.xml7.1.5.2hodge-theory-iii-7.1.5 \mathrm {R}{a_\bullet }_* \colon \mathrm {D}^+\mathrm {F}_2(X_\bullet ) \to \mathrm {D}^+\mathrm {F}_2(S_\bullet ). \tag{7.1.5.2} 2362hodge-theory-iii-7.1.6hodge-theory-iii-7.1.6.xml7.1.6hodge-theory-iii-7.1
Let K be a complex of cosimplicial sheaves on S, endowed with an increasing filtration W.
We define the diagonal filtration \delta (W,L) of W and L to be the increasing filtration of \mathrm {s} K given by
2357Equationhodge-theory-iii-7.1.6.1hodge-theory-iii-7.1.6.1.xml7.1.6.1hodge-theory-iii-7.1.6 \begin {aligned} \delta (W,L)_n(\mathrm {s} K) &= \bigoplus _{p,q} W_{n+p}(K^{p,q}) \\&= \sum _p \mathrm {s}(W_{n+p}(K)) \cap L^p(\mathrm {s} K). \end {aligned} \tag{7.1.6.1}
We have
2358Equationhodge-theory-iii-7.1.6.2hodge-theory-iii-7.1.6.2.xml7.1.6.2hodge-theory-iii-7.1.6 \operatorname {Gr}_n^{\delta (W,L)}(\mathrm {s} K) \simeq \bigoplus _p \operatorname {Gr}_{n+p}^W(K^{\bullet p})[-p]. \tag{7.1.6.2}
The functor (K,W)\mapsto (\mathrm {s} K,\delta (W,L)) sends filtered quasi-isomorphisms to filtered quasi-isomorphisms, and defines
2359Equationhodge-theory-iii-7.1.6.3hodge-theory-iii-7.1.6.3.xml7.1.6.3hodge-theory-iii-7.1.6 \begin {aligned} (s,\delta ) \colon \mathrm {D}^+\mathrm {F}(S_\bullet ) &\to \mathrm {D}^+\mathrm {F}(S) \\(K,W) &\mapsto (\mathrm {s} K,\delta (W,L)) \end {aligned} \tag{7.1.6.3}
whence, by composition with ,
2360Equationhodge-theory-iii-7.1.6.4hodge-theory-iii-7.1.6.4.xml7.1.6.4hodge-theory-iii-7.1.6 \begin {aligned} (\mathrm {R}\Gamma ,\delta (-,L)) \colon \mathrm {D}^+\mathrm {F}(X_\bullet ) &\to \mathrm {D}^+\mathrm {F}(S) \\(K,W) &\mapsto (\mathrm {R}\Gamma K,\delta (W,L)). \end {aligned} \tag{7.1.6.4}
From , we see that
2361Equationhodge-theory-iii-7.1.6.5hodge-theory-iii-7.1.6.5.xml7.1.6.5hodge-theory-iii-7.1.6 \operatorname {Gr}_n^{\delta (W,L)} = \bigoplus _p \mathrm {R}{a_p}_*(\operatorname {Gr}_{n+p}^W K)[-p]. \tag{7.1.6.5} 2366hodge-theory-iii-7.1.7hodge-theory-iii-7.1.7.xml7.1.7hodge-theory-iii-7.1
If (K,W,F) is a bifiltered complex of cosimplicial sheaves, then \mathrm {s} K is endowed with the three filtrations W, F, and L, and defines different bifiltered complexes.
For example, for increasing W, the functor (K,W,F)\mapsto (K,\delta (W,L),F) sends bifiltered quasi-isomorphisms to bifiltered quasi-isomorphisms, and thus defines
2363Equationhodge-theory-iii-7.1.7.1hodge-theory-iii-7.1.7.1.xml7.1.7.1hodge-theory-iii-7.1.7 \begin {aligned} \mathrm {D}^+\mathrm {F}_2(S_\bullet ) &\to \mathrm {D}^+\mathrm {F}_2(S) \\(K,W,F) &\mapsto (K,\delta (W,L),F). \end {aligned} \tag{7.1.7.1}
By composition with , we thus obtain
2364Equationhodge-theory-iii-7.1.7.2hodge-theory-iii-7.1.7.2.xml7.1.7.2hodge-theory-iii-7.1.7 \begin {aligned} \mathrm {D}^+\mathrm {F}_2(X_\bullet ) &\to \mathrm {D}^+\mathrm {F}_2(S) \\(K,W,F) &\mapsto (\mathrm {R}\Gamma K,\delta (W,L),F). \end {aligned} \tag{7.1.7.2}
and we have that
2365Equationhodge-theory-iii-7.1.7.3hodge-theory-iii-7.1.7.3.xml7.1.7.3hodge-theory-iii-7.1.7 \operatorname {Gr}_n^{\delta (W,L)}(\mathrm {R}\Gamma K,F) = \bigoplus _p \mathrm {R}{a_p}_*(\operatorname {Gr}_{n+p}^W K,F)[-p] \tag{7.1.7.3}
in \mathrm {D}^+\mathrm {F}(S).
2774hodge-theory-iii-7.2hodge-theory-iii-7.2.xmlSupplements to the two filtrations lemma7.2hodge-theory-iii-7
In this section, we give a new proof of the two filtrations lemma () and some supplements.
961hodge-theory-iii-7.2.1hodge-theory-iii-7.2.1.xml7.2.1hodge-theory-iii-7.2
Let (K,F,W) be a bounded below bifiltered complex with objects in an abelian category \mathscr {A}, with F assumed to be biregular.
We say that (K,F,W) is F-splitable if the filtered complex (K,W) can be written as a sum of filtered complexes
(K,W) = \bigoplus _{n\in \mathbb {Z}} (K_n,W_n)
with
F^nK = \bigoplus _{n'\geqslant n} K_{n'}.
Let r_0\geqslant 0 be an integer or +\infty .
The following condition was considered in and :
962Conditionhodge-theory-iii-7.2.2hodge-theory-iii-7.2.2.xml7.2.2hodge-theory-iii-7.2
For every non-negative integer r<r_0, the differentials d_r of the graded complex E_r(K,W) are strictly compatible with the recurrent filtration defined by F.
963hodge-theory-iii-7.2.3hodge-theory-iii-7.2.3.xml7.2.3hodge-theory-iii-7.2
It is clear that, if (K,F,W) is F-splitable, then is satisfied for r_0=\infty .
Conversely, it seems that if is satisfied for r_0=\infty , then everything is as if the functor \operatorname {Gr}_F were exact.
For example, we will show that the \operatorname {Gr}_F of the spectral sequence E(K,W) can then be identified with the spectral sequence E(\operatorname {Gr}_FK,W), and that the spectral sequence E(K,F) degenerates (E_1=E_\infty ).
964hodge-theory-iii-7.2.4hodge-theory-iii-7.2.4.xml7.2.4hodge-theory-iii-7.2
We immediately deduce from the definition () that the first direct filtration of E_r(K,W) is the filtration F_d of E_r(K,W) by the images
F_d^p(E_r(K,W)) = \operatorname {Im} \big (E_r(F^pK,W) \to E_r(K,W)\big ).
Dually, the second direct filtration () of E_r(K,W) is the filtration F_{d^*} of E_r(K,W) by the kernels
F_{d^*}^p(E_r(K,W)) = \operatorname {Ker} \big (E_r(K,W) \to E_r(K/F^pK,W)\big ).
The recurrent filtration F_{\mathrm {rec}} of E_r(K,W) is intermediary between these two filtrations ((iii) of ).
969Propositionhodge-theory-iii-7.2.5hodge-theory-iii-7.2.5.xml7.2.5hodge-theory-iii-7.2
Suppose that (K,F,W) satisfies for some r_0\geqslant 0.
Then
(F^aK/F^bK,F,W) also satisfies for r_0.
For r\leqslant r_0, the sequence
0 \to E_r(F^pK,W) \to E_r(K,W) \to E_r(K/F^pK,W) \to 0
is exact;
for r=r_0+1, the sequence
E_r(F^pK,W) \to E_r(K,W) \to E_r(K/F^pK,W)
is exact.
In particular, for r\leqslant r_0+1, the two direct filtrations and the recurrent filtration of E_r(K,W) agree.
968Proof#176unstable-176.xmlhodge-theory-iii-7.2.5
Fix
3019hodge-theory-iii-terminology-and-notationhodge-theory-iii-terminology-and-notation.xmlHodge Theory III › Terminology and notationhodge-theory-iii1912hodge-theory-iii-0.1hodge-theory-iii-0.1.xmlIII.0.1hodge-theory-iii-terminology-and-notation
Let u\colon X\to Y be a continuous map between topological spaces.
We say that u is proper if it is proper in the sense of Bourbaki (i.e. universally closed) and furthermore separated (i.e. the diagonal of X\times _Y X is closed).
1913hodge-theory-iii-0.2hodge-theory-iii-0.2.xmlIII.0.2hodge-theory-iii-terminology-and-notation
Following Gabriel and Zisman, we say simplicial where we would have previously said semi-simplicial.
1914hodge-theory-iii-0.3hodge-theory-iii-0.3.xmlIII.0.3hodge-theory-iii-terminology-and-notation
We denote by A a Noetherian subring of \mathbb {R} such that A\otimes \mathbb {Q} is a field.
The useful cases are A=\mathbb {Z}, \mathbb {Q}, or \mathbb {R}.
1915hodge-theory-iii-0.4hodge-theory-iii-0.4.xmlIII.0.4hodge-theory-iii-terminology-and-notation
A mixed Hodge A-structure consists of an A-module H_A of finite type, a finite increasing filtration W on the A\otimes \mathbb {Q}-vector space H_{A\otimes \mathbb {Q}}=H_A\otimes \mathbb {Q}, and a finite decreasing filtration F on the \mathbb {C}-vector space H_\mathbb {C}=H_A\otimes _A\mathbb {C}.
We demand that the (\operatorname {Gr}_n^W(H_{A\otimes \mathbb {Q}}),\operatorname {Gr}_n^W(F)) be Hodge A\otimes \mathbb {Q}-structures.
For A=\mathbb {Z} (resp. A=\mathbb {Q}), we recover (resp. );
the results of carry over as they are.
1916hodge-theory-iii-0.5hodge-theory-iii-0.5.xmlIII.0.5hodge-theory-iii-terminology-and-notation
Let \mathscr {E} be a subset of \mathbb {Z}\times \mathbb {Z}.
We say that a mixed Hodge structure H is of degree \mathscr {E} if the Hodge numbers h^{pq} are zero for (p,q)\not \in \mathscr {E}.
1917hodge-theory-iii-0.6hodge-theory-iii-0.6.xmlIII.0.6hodge-theory-iii-terminology-and-notation
From now on, we denote by \Omega _X^\bullet (\log D) what we previously denoted by \Omega _X^\bullet \langle D\rangle ().
1918hodge-theory-iii-0.7hodge-theory-iii-0.7.xmlIII.0.7hodge-theory-iii-terminology-and-notation
Unless explicitly stated otherwise, scheme means "scheme of finite type over \mathbb {C}", and a sheaf on a scheme X is a sheaf on the underlying topological space of X_\mathrm {an}.
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