1.4 Hypercohomology of filtered complexes
In this section, we recall some standard constructions in hypercohomology. We do not use the language of derived categories, which would be more natural here.
Throughout this entire section, by “complex” we mean “bounded-below complex”.
Let T be a left-exact functor from an abelian category \mathscr{A} to an abelian category \mathscr{B}. Suppose that every object of \mathscr{A} injects into an injective object; the derived functors \mathrm{R}^iT\colon\mathscr{A}\to\mathscr{B} are then defined. An object A of \mathscr{A} is said to be acyclic for T if \mathrm{R}^iT(A)=0 for i>0.
Let (A,F) be a filtered object with finite filtration, and TF the filtration of TA by its sub-objects TF^p(A) (these are sub-objects since T is left exact). If \operatorname{Gr}_F(A) is T-acyclic, then the F^p(A) are T-acyclic as successive extensions of T-acyclic objects. The image under T of the sequence 0 \to F^{p+1}(A) \to F^p(A) \to \operatorname{Gr}^p(A) \to 0 is thus exact, and \operatorname{Gr}_{FT}TA \xrightarrow{\sim}T\operatorname{Gr}_FA. \tag{1.4.2.1}
Let A be an object endowed with finite filtrations F and W such that \operatorname{Gr}_F\operatorname{Gr}_W A are T-acyclic. The objects \operatorname{Gr}_FA and \operatorname{Gr}_WA are then T-acyclic, as well as the F^q(A)\cap W^p(A). The sequences 0 \to T(F^q\cap W^{p+1}) \to T(F^q\cap W^p) \to T((F^q\cap W^p)/(F^q\cap W^{p+1})) \to 0 are thus exact, and T(F^q(\operatorname{Gr}_W^p(A))) is the image in T(\operatorname{Gr}_W^p(A)) of T(F^p\cap W^q). The diagram \begin{CD} T(F^q\cap W^p) @>>> T(F^q\operatorname{Gr}_W^pA) @>>> T\operatorname{Gr}_W^pA \\@V{\cong}VV @. @VV{\cong}V \\TF^q\cap TW^p @= TF^q\cap TW^p @>>> \operatorname{Gr}_{TW}^pTA \end{CD} then shows that the isomorphism in (1.4.2.1) relative to W sends the filtration \operatorname{Gr}_{TW}(TF) to the filtration T(\operatorname{Gr}_W(F)).
Let K be a complex of objects of \mathscr{A}. The hypercohomology objects \mathrm{R}^iT(K) are calculated as follows:
- We choose a quasi-isomorphism i\colon K\to K such that the components of K' are acyclic for T. For example, we can take K' to be the simple complex associated to an injective Cartan–Eilenberg resolution of K.
- We set \mathrm{R}^iT(K) = \operatorname{H}^i(T(K')).
We can show that \mathrm{R}^iT(K) does not depend on the choice of K', but depends functorially on K, and that a quasi-isomorphism f\colon K_1\to K_2 induces isomorphisms \mathrm{R}^iT(f)\colon \mathrm{R}^iT(K_1) \to \mathrm{R}^iT(K_2).
Let F be a biregular filtration of K. A
The differentials d_1 of this spectral sequence are the connection morphisms defined by the short exact sequences 0 \to \operatorname{Gr}^{p+1}K \to F^pK/F^{p+2}K \to \operatorname{Gr}^pK \to 0.
Let K be a complex. We denote by \tau_{\leqslant p}(K) the following subcomplex: \tau_{\leqslant p}(K)^n = \begin{cases} K^n &\text{for }n<p \\\operatorname{Ker}(d) &\text{for }n=p \\0 &\text{for }n>p. \end{cases}
The filtration, said to be canonical, of K by the \tau_{\leqslant p}(K) is induced by shifting the trivial filtration G for which G^0(K)=K and G^1(K)=0. We have, for the canonical filtration, \begin{aligned} E_1^{pq} = 0 &\qquad\text{if }p+q\neq-p \\\\H^{-p} &\qquad\text{if }p+q=-p. \end{aligned} A quasi-isomorphism f\colon K\to K' is automatically a filtered quasi-isomorphism for the canonical filtrations.
The subcomplexes \sigma_{\geqslant p}(K) of K \sigma_{\geqslant p}(K)^n = \begin{cases} 0 &\text{if }n<p \\K^n &\text{if }n\geqslant p \end{cases} define a biregular filtration, called the stupid filtration of K.
The hypercohomology spectral sequences attached to the stupid or canonical filtrations of K are the two hypercohomology spectral sequences of K.
Let f\colon X\to Y be a continuous map between topological spaces, and let \mathscr{F} be an abelian sheaf on X. Let \mathscr{F}^\bullet be a resolution of \mathscr{F} by f_*-acyclic sheaves. We have \mathrm{R}^i f_*\mathscr{F}\simeq\operatorname{\mathscr{H}}^i(f_*\mathscr{F}^\bullet). We take the functor T to be the functor \Gamma(Y,-). The hypercohomology spectral sequence of the complex f_*\mathscr{F}^\bullet endowed with its canonical filtration (1.4.6) E_1^{pq} = \operatorname{H}^{2p+q}(Y,\mathrm{R}^{-p}f_*\mathscr{F}) \Rightarrow\operatorname{H}^{p+q}(X,\mathscr{F}) is exactly, up to the renumbering E_r^{pq}\mapsto E_{r+1}^{2p+q,-p}, the Leray spectral sequence for f and \mathscr{F}.
Let (K,W,F) be a biregular bifiltered complex. To this complex, we associate:
- A spectral sequence {}_W E_1^{p,n-p} = \operatorname{H}^n(\operatorname{Gr}_W^p(K)) \Rightarrow\operatorname{H}^n(K) with differentials {}_W d_1 being the connecting morphisms induced by the short exact sequences 0 \to \operatorname{Gr}_W^{p+1}(K) \to W^p(K)/W^{p+2}(K) \to \operatorname{Gr}_W^p(K) \to 0;
- An analogous spectral sequence for the filtration F;
- Exact squares
The exterior rows and columns of this square define connection morphisms \begin{aligned} {}_{F,W}d_1\colon \operatorname{H}^n\operatorname{Gr}_F^p\operatorname{Gr}_W^q K &\to \operatorname{H}^{n+1}\operatorname{Gr}_F^{p+1}\operatorname{Gr}_W^q K \\{}_{W,F}d_1\colon \operatorname{H}^n\operatorname{Gr}_F^p\operatorname{Gr}_W^q K &\to \operatorname{H}^{n+1}\operatorname{Gr}_F^p\operatorname{Gr}_W^{q+1} K. \end{aligned} These morphisms satisfy \begin{gathered} {}_{FW}d_1 \circ {}_{WF}d_1 + {}_{WF}d_1 \circ {}_{FW}d_1 = 0 \\ {}_{FW}d_1^2 = 0 \\ {}_{WF}d_1^2 = 0 \end{gathered}
The morphisms {}_{FW}d_1 are the morphisms d_1 of the spectral sequences E^{(q)}, with the E_1^{(q)p,n-p} term equal to E_1^{p,q,n-p-q} \coloneqq \operatorname{H}^n(\operatorname{Gr}_F^p\operatorname{Gr}_W^q K) \Rightarrow\operatorname{H}^n(\operatorname{Gr}_W^q K) = {}_WE_1^{q,n-q} \tag{1.4.9.1} defined by the filtered complex \operatorname{Gr}_W^q(K). This spectral sequence abuts to the filtration induced by F on \operatorname{H}^\bullet\operatorname{Gr}_W^q K. Similarly, the {}_{WF}d_1 are the d_1 of spectral sequences with the same initial terms E_1^{p,q,n-p-q} \coloneqq \operatorname{H}^n(\operatorname{Gr}_F^p\operatorname{Gr}_W^q K) \Rightarrow\operatorname{H}^n(\operatorname{Gr}_F^p K). \tag{1.4.9.1}
These constructions are symmetric in F and W via the isomorphism \operatorname{Gr}_F^p\operatorname{Gr}_W^q \sim \operatorname{Gr}_W^q\operatorname{Gr}_F^p.
We can also interpret the {}_{W,F}d_1 are the initial morphisms of a morphism of spectral sequences (1.4.9.2), abutting to {}_Wd_1.
Indeed, let C^q be the cone of the morphism W^q(K)/W^{q+2}(K) \to \operatorname{Gr}_W^q(K). In the diagram \Sigma\colon \operatorname{Gr}_W^{q+1}(K)[1] \xrightarrow{u} C^q \xleftarrow{i} \operatorname{Gr}_W^q(K) the morphism u is a quasi-isomorphism, and we have {}_Wd_1 = \operatorname{H}(u)^{-1}\circ\operatorname{H}(i).
In fact, u is even a filtered (for F) quasi-isomorphism, and the above construction defines a morphism from the spectral sequence defined by (\operatorname{Gr}_W^q(K),F) to that defined by (\operatorname{Gr}_W^{q+1}(K)[1],F), and it abuts to {}_Wd_1. The initial term of this morphism, induced by \operatorname{Gr}_F(\Sigma) is exactly {}_{W,F}d_1.
These constructions pass as they are to hypercohomology. Let K be a complex endowed with two biregular filtrations F and W.
A bifiltered T-acyclic resolution of K is a bifiltered quasi-isomorphism i\colon K\to K' such that the \operatorname{Gr}_F^p\operatorname{Gr}_W^n({K'}^m) are T-acyclic. Such a morphism always exists. In the particular case where \mathscr{A} is the category of sheaves of A-modules on a topological space X, and where T is the functor \Gamma from \mathscr{A} to the category of A-modules, then an example of a bifiltered T-acyclic resolution of K is the simple complex associated to the double complex given by the Godement resolution \mathscr{C}^\bullet(K) of K, filtered by the \mathscr{C}^\bullet(F^p(K)) and the \mathscr{C}^\bullet(W^n(K)). Since \mathscr{C}^\bullet is exact, we have \operatorname{Gr}_F\operatorname{Gr}_W(\mathscr{C}^\bullet(K)) \simeq \mathscr{C}^\bullet(\operatorname{Gr}_F\operatorname{Gr}_W(K)). We will not have need here of any other case.
If K' is a bifiltered T-acyclic resolution of K, then the complex TK' is filtered by the TF^pK' and the TW^qK' ((1.4.3)). Furthermore, \operatorname{Gr}_W^n(K') is a T-acyclic filtered (for F) resolution of \operatorname{Gr}_F^n(K), and \operatorname{Gr}_F^n\operatorname{Gr}_W^m(K') is a T-acyclic resolution of \operatorname{Gr}_F^n\operatorname{Gr}_W^m(K), and \begin{gathered} T\operatorname{Gr}_F K' \approx \operatorname{Gr}_F TK' \qquad\text{as }W\text{-filtered complexes} \\T\operatorname{Gr}_W K' \approx \operatorname{Gr}_W TK' \qquad\text{as }F\text{-filtered complexes} \\T\operatorname{Gr}_F\operatorname{Gr}_W K' \approx \operatorname{Gr}_F\operatorname{Gr}_W TK'. \end{gathered}
Under the hypotheses of (1.4.11):
- The initial terms of the hypercohomology spectral sequences {}_WE_1^{q,n-q} = \mathrm{R}^nT(\operatorname{Gr}_W^qK) \Rightarrow\mathrm{R}^nT(K) \tag{1} {}_FE_1^{p,n-p} = \mathrm{R}^nT(\operatorname{Gr}_F^pK) \Rightarrow\mathrm{R}^nT(K) \tag{2} are abutments of the hypercohomology spectral sequences of the filtered complexes \operatorname{Gr}_W^qK and \operatorname{Gr}_F^pK, with E_1 pages given by E_1^{p,q,n-p-q} \coloneqq \mathrm{R}^nT(\operatorname{Gr}_F^p\operatorname{Gr}_W^qK) \Rightarrow{}_WE_1^{q,n-q} \qquad\text{for fixed }q \tag{3} E_1^{p,q,n-p-q} \coloneqq \mathrm{R}^nT(\operatorname{Gr}_F^p\operatorname{Gr}_W^qK) \Rightarrow{}_FE_1^{p,n-p} \qquad\text{for fixed }p \tag{4}
- The filtration of {}_WE_1^{p,n-p}, abutting to the spectral sequence (3), is the filtration of {}_WE_1^{p,n-p}(TK') induced by the filtration F of TK'.
- For the differentials of the complexes \operatorname{Gr}_W^n(T(K')) to be strictly compatible with the filtration F, it is necessary and sufficient that the hypercohomology spectral sequences (3) degenerate on the E_1 page.
- The morphisms d_1 of the spectral sequence (4) are the initial terms of the degree-1 morphisms of the spectral sequences (3) that abut to the morphisms d_1 of the spectral sequence (1).