7.1 Filtered derived category
This section completes §1.4.
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 (1.3.6); this is the derived filtered category (resp. derived bifiltered category).
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 (1.4.9).
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 \mathrm{R}\colon \mathrm{D}^+(\mathscr{A}) \to \mathrm{D}^+(\mathscr{B}) \tag{7.1.3.1} \mathrm{R}\colon \mathrm{D}^+\mathrm{F}(\mathscr{A}) \to \mathrm{D}^+\mathrm{F}(\mathscr{B}) \tag{7.1.3.2} \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 ((1.4.5) and (1.4.11)), 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. (1.4.5)).
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 (5.1.12), and recall (5.2.6.1): \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 (5.1.9.3). 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 (5.2.7.1).
If K is filtered (resp. bifiltered), then \mathrm{R}{a_\bullet}_*K is filtered (resp. bifiltered): we have \mathrm{R}{a_\bullet}_* \colon \mathrm{D}^+\mathrm{F}(X_\bullet) \to \mathrm{D}^+\mathrm{F}(S_\bullet) \tag{7.1.5.1} \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}
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 \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 \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 \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 (7.1.5.1), \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 (7.1.6.2), we see that \operatorname{Gr}_n^{\delta(W,L)} = \bigoplus_p \mathrm{R}{a_p}_*(\operatorname{Gr}_{n+p}^W K)[-p]. \tag{7.1.6.5}
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 \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 (7.1.5.2), we thus obtain \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 \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).