Definition 13.13.1. Let $\mathcal{A}$ be an abelian category. The category of finite filtered objects of $\mathcal{A}$ is the category of filtered objects $(A, F)$ of $\mathcal{A}$ whose filtration $F$ is finite. We denote it $\text{Fil}^ f(\mathcal{A})$.
13.13 Filtered derived categories
A reference for this section is [I, Chapter V, cotangent]. Let $\mathcal{A}$ be an abelian category. In this section we will define the filtered derived category $DF(\mathcal{A})$ of $\mathcal{A}$. In short, we will define it as the derived category of the exact category of objects of $\mathcal{A}$ endowed with a finite filtration. (Thus our construction is a special case of a more general construction of the derived category of an exact category, see for example [Buhler], [Keller].) Illusie's filtered derived category is the full subcategory of ours consisting of those objects whose filtration is finite. (In our category the filtration is still finite in each degree, but may not be uniformly bounded.) The rationale for our choice is that it is not harder and it allows us to apply the discussion to the spectral sequences of Lemma 13.21.3, see also Remark 13.21.4.
We will use the notation regarding filtered objects introduced in Homology, Section 12.19. The category of filtered objects of $\mathcal{A}$ is denoted $\text{Fil}(\mathcal{A})$. All filtrations will be decreasing by fiat.
Thus $\text{Fil}^ f(\mathcal{A})$ is a full subcategory of $\text{Fil}(\mathcal{A})$. For each $p \in \mathbf{Z}$ there is a functor $\text{gr}^ p : \text{Fil}^ f(\mathcal{A}) \to \mathcal{A}$. There is a functor
\[ \text{gr} = \bigoplus \nolimits _{p \in \mathbf{Z}} \text{gr}^ p : \text{Fil}^ f(\mathcal{A}) \to \text{Gr}(\mathcal{A}) \]
where $\text{Gr}(\mathcal{A})$ is the category of graded objects of $\mathcal{A}$, see Homology, Definition 12.16.1. Finally, there is a functor
\[ (\text{forget }F) : \text{Fil}^ f(\mathcal{A}) \longrightarrow \mathcal{A} \]
which associates to the filtered object $(A, F)$ the underlying object of $\mathcal{A}$. The category $\text{Fil}^ f(\mathcal{A})$ is an additive category, but not abelian in general, see Homology, Example 12.3.13.
Because the functors $\text{gr}^ p$, $\text{gr}$, $(\text{forget }F)$ are additive they induce exact functors of triangulated categories
\[ \text{gr}^ p, (\text{forget }F) : K(\text{Fil}^ f(\mathcal{A})) \to K(\mathcal{A}) \quad \text{and}\quad \text{gr} : K(\text{Fil}^ f(\mathcal{A})) \to K(\text{Gr}(\mathcal{A})) \]
by Lemma 13.10.6. By analogy with the case of the homotopy category of an abelian category we make the following definitions.
Definition 13.13.2. Let $\mathcal{A}$ be an abelian category.
Let $\alpha : K^\bullet \to L^\bullet $ be a morphism of $K(\text{Fil}^ f(\mathcal{A}))$. We say that $\alpha $ is a filtered quasi-isomorphism if the morphism $\text{gr}(\alpha )$ is a quasi-isomorphism.
Let $K^\bullet $ be an object of $K(\text{Fil}^ f(\mathcal{A}))$. We say that $K^\bullet $ is filtered acyclic if the complex $\text{gr}(K^\bullet )$ is acyclic.
Note that $\alpha : K^\bullet \to L^\bullet $ is a filtered quasi-isomorphism if and only if each $\text{gr}^ p(\alpha )$ is a quasi-isomorphism. Similarly a complex $K^\bullet $ is filtered acyclic if and only if each $\text{gr}^ p(K^\bullet )$ is acyclic.
Lemma 13.13.3. Let $\mathcal{A}$ be an abelian category.
The functor $K(\text{Fil}^ f(\mathcal{A})) \longrightarrow \text{Gr}(\mathcal{A})$, $K^\bullet \longmapsto H^0(\text{gr}(K^\bullet ))$ is homological.
The functor $K(\text{Fil}^ f(\mathcal{A})) \rightarrow \mathcal{A}$, $K^\bullet \longmapsto H^0(\text{gr}^ p(K^\bullet ))$ is homological.
The functor $K(\text{Fil}^ f(\mathcal{A})) \longrightarrow \mathcal{A}$, $K^\bullet \longmapsto H^0((\text{forget }F)K^\bullet )$ is homological.
Proof. This follows from the fact that $H^0 : K(\mathcal{A}) \to \mathcal{A}$ is homological, see Lemma 13.11.1 and the fact that the functors $\text{gr}, \text{gr}^ p, (\text{forget }F)$ are exact functors of triangulated categories. See Lemma 13.4.20. $\square$
Lemma 13.13.4. Let $\mathcal{A}$ be an abelian category. The full subcategory $\text{FAc}(\mathcal{A})$ of $K(\text{Fil}^ f(\mathcal{A}))$ consisting of filtered acyclic complexes is a strictly full saturated triangulated subcategory of $K(\text{Fil}^ f(\mathcal{A}))$. The corresponding saturated multiplicative system (see Lemma 13.6.10) of $K(\text{Fil}^ f(\mathcal{A}))$ is the set $\text{FQis}(\mathcal{A})$ of filtered quasi-isomorphisms. In particular, the kernel of the localization functor is $\text{FAc}(\mathcal{A})$ and the functor $H^0 \circ \text{gr}$ factors through $Q$.
\[ Q : K(\text{Fil}^ f(\mathcal{A})) \longrightarrow \text{FQis}(\mathcal{A})^{-1}K(\text{Fil}^ f(\mathcal{A})) \]
Proof. We know that $H^0 \circ \text{gr}$ is a homological functor by Lemma 13.13.3. Thus this lemma is a special case of Lemma 13.6.11. $\square$
Definition 13.13.5. Let $\mathcal{A}$ be an abelian category. Let $\text{FAc}(\mathcal{A})$ and $\text{FQis}(\mathcal{A})$ be as in Lemma 13.13.4. The filtered derived category of $\mathcal{A}$ is the triangulated category
\[ DF(\mathcal{A}) = K(\text{Fil}^ f(\mathcal{A}))/\text{FAc}(\mathcal{A}) = \text{FQis}(\mathcal{A})^{-1} K(\text{Fil}^ f(\mathcal{A})). \]
Lemma 13.13.6. The functors $\text{gr}^ p, \text{gr}, (\text{forget }F)$ induce canonical exact functors which commute with the localization functors.
\[ \text{gr}^ p, \text{gr}, (\text{forget }F): DF(\mathcal{A}) \longrightarrow D(\mathcal{A}) \]
Proof. This follows from the universal property of localization, see Lemma 13.5.7, provided we can show that a filtered quasi-isomorphism is turned into a quasi-isomorphism by each of the functors $\text{gr}^ p, \text{gr}, (\text{forget }F)$. This is true by definition for the first two. For the last one the statement we have to do a little bit of work. Let $f : K^\bullet \to L^\bullet $ be a filtered quasi-isomorphism in $K(\text{Fil}^ f(\mathcal{A}))$. Choose a distinguished triangle $(K^\bullet , L^\bullet , M^\bullet , f, g, h)$ which contains $f$. Then $M^\bullet $ is filtered acyclic, see Lemma 13.13.4. Hence by the corresponding lemma for $K(\mathcal{A})$ it suffices to show that a filtered acyclic complex is an acyclic complex if we forget the filtration. This follows from Homology, Lemma 12.19.15. $\square$
Definition 13.13.7. Let $\mathcal{A}$ be an abelian category. The bounded filtered derived category $DF^ b(\mathcal{A})$ is the full subcategory of $DF(\mathcal{A})$ with objects those $X$ such that $\text{gr}(X) \in D^ b(\mathcal{A})$. Similarly for the bounded below filtered derived category $DF^{+}(\mathcal{A})$ and the bounded above filtered derived category $DF^{-}(\mathcal{A})$.
Lemma 13.13.8. Let $\mathcal{A}$ be an abelian category. Let $K^\bullet \in K(\text{Fil}^ f(\mathcal{A}))$.
If $H^ n(\text{gr}(K^\bullet )) = 0$ for all $n < a$, then there exists a filtered quasi-isomorphism $K^\bullet \to L^\bullet $ with $L^ n = 0$ for all $n < a$.
If $H^ n(\text{gr}(K^\bullet )) = 0$ for all $n > b$, then there exists a filtered quasi-isomorphism $M^\bullet \to K^\bullet $ with $M^ n = 0$ for all $n > b$.
If $H^ n(\text{gr}(K^\bullet )) = 0$ for all $|n| \gg 0$, then there exists a commutative diagram of morphisms of complexes
where all the arrows are filtered quasi-isomorphisms, $L^\bullet $ bounded below, $M^\bullet $ bounded above, and $N^\bullet $ a bounded complex.
\[ \xymatrix{ K^\bullet \ar[r] & L^\bullet \\ M^\bullet \ar[u] \ar[r] & N^\bullet \ar[u] } \]
Proof. Suppose that $H^ n(\text{gr}(K^\bullet )) = 0$ for all $n < a$. By Homology, Lemma 12.19.15 the sequence
\[ K^{a - 1} \xrightarrow {d^{a - 2}} K^{a - 1} \xrightarrow {d^{a - 1}} K^ a \]
is an exact sequence of objects of $\mathcal{A}$ and the morphisms $d^{a - 2}$ and $d^{a - 1}$ are strict. Hence $\mathop{\mathrm{Coim}}(d^{a - 1}) = \mathop{\mathrm{Im}}(d^{a - 1})$ in $\text{Fil}^ f(\mathcal{A})$ and the map $\text{gr}(\mathop{\mathrm{Im}}(d^{a - 1})) \to \text{gr}(K^ a)$ is injective with image equal to the image of $\text{gr}(K^{a - 1}) \to \text{gr}(K^ a)$, see Homology, Lemma 12.19.13. This means that the map $K^\bullet \to \tau _{\geq a}K^\bullet $ into the truncation
\[ \tau _{\geq a}K^\bullet = (\ldots \to 0 \to K^ a/\mathop{\mathrm{Im}}(d^{a - 1}) \to K^{a + 1} \to \ldots ) \]
is a filtered quasi-isomorphism. This proves (1). The proof of (2) is dual to the proof of (1). Part (3) follows formally from (1) and (2). $\square$
To state the following lemma denote $\text{FAc}^{+}(\mathcal{A})$, $\text{FAc}^{-}(\mathcal{A})$, resp. $\text{FAc}^ b(\mathcal{A})$ the intersection of $K^{+}(\text{Fil}^ f\mathcal{A})$, $K^{-}(\text{Fil}^ f\mathcal{A})$, resp. $K^ b(\text{Fil}^ f\mathcal{A})$ with $\text{FAc}(\mathcal{A})$. Denote $\text{FQis}^{+}(\mathcal{A})$, $\text{FQis}^{-}(\mathcal{A})$, resp. $\text{FQis}^ b(\mathcal{A})$ the intersection of $K^{+}(\text{Fil}^ f\mathcal{A})$, $K^{-}(\text{Fil}^ f\mathcal{A})$, resp. $K^ b(\text{Fil}^ f\mathcal{A})$ with $\text{FQis}(\mathcal{A})$.
Lemma 13.13.9. Let $\mathcal{A}$ be an abelian category. The subcategories $\text{FAc}^{+}(\mathcal{A})$, $\text{FAc}^{-}(\mathcal{A})$, resp. $\text{FAc}^ b(\mathcal{A})$ are strictly full saturated triangulated subcategories of $K^{+}(\text{Fil}^ f\mathcal{A})$, $K^{-}(\text{Fil}^ f\mathcal{A})$, resp. $K^ b(\text{Fil}^ f\mathcal{A})$. The corresponding saturated multiplicative systems (see Lemma 13.6.10) are the sets $\text{FQis}^{+}(\mathcal{A})$, $\text{FQis}^{-}(\mathcal{A})$, resp. $\text{FQis}^ b(\mathcal{A})$.
The kernel of the functor $K^{+}(\text{Fil}^ f\mathcal{A}) \to DF^{+}(\mathcal{A})$ is $\text{FAc}^{+}(\mathcal{A})$ and this induces an equivalence of triangulated categories
The kernel of the functor $K^{-}(\text{Fil}^ f\mathcal{A}) \to DF^{-}(\mathcal{A})$ is $\text{FAc}^{-}(\mathcal{A})$ and this induces an equivalence of triangulated categories
The kernel of the functor $K^ b(\text{Fil}^ f\mathcal{A}) \to DF^ b(\mathcal{A})$ is $\text{FAc}^ b(\mathcal{A})$ and this induces an equivalence of triangulated categories
\[ K^{+}(\text{Fil}^ f\mathcal{A})/\text{FAc}^{+}(\mathcal{A}) = \text{FQis}^{+}(\mathcal{A})^{-1}K^{+}(\text{Fil}^ f\mathcal{A}) \longrightarrow DF^{+}(\mathcal{A}) \]
\[ K^{-}(\text{Fil}^ f\mathcal{A})/\text{FAc}^{-}(\mathcal{A}) = \text{FQis}^{-}(\mathcal{A})^{-1}K^{-}(\text{Fil}^ f\mathcal{A}) \longrightarrow DF^{-}(\mathcal{A}) \]
\[ K^ b(\text{Fil}^ f\mathcal{A})/\text{FAc}^ b(\mathcal{A}) = \text{FQis}^ b(\mathcal{A})^{-1}K^ b(\text{Fil}^ f\mathcal{A}) \longrightarrow DF^ b(\mathcal{A}) \]
Proof. This follows from the results above, in particular Lemma 13.13.8, by exactly the same arguments as used in the proof of Lemma 13.11.6. $\square$
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