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13.14. Filtered derived categories

A reference for this section is [cotangent, I, Chapter V]. 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 12.16. The category of filtered objects of $\mathcal{A}$ is denoted $\text{Fil}(\mathcal{A})$. All filtrations will be decreasing by fiat.

Definition 13.14.1. 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 $\text{Fil}^f(\mathcal{A})$.

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 12.15.1. 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 13.14.2. Let $\mathcal{A}$ be an abelian category.

  1. 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.
  2. 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 13.14.3. Let $\mathcal{A}$ be an abelian category.

  1. The functor $K(\text{Fil}^f(\mathcal{A})) \longrightarrow \text{Gr}(\mathcal{A})$, $K^\bullet \longmapsto H^0(\text{gr}(K^\bullet))$ is homological.
  2. The functor $K(\text{Fil}^f(\mathcal{A})) \rightarrow \mathcal{A}$, $K^\bullet \longmapsto H^0(\text{gr}^p(K^\bullet))$ is homological.
  3. 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 13.4.19. $\square$

Lemma 13.14.4. 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 $$ Q : K(\text{Fil}^f(\mathcal{A})) \longrightarrow \text{FQis}(\mathcal{A})^{-1}K(\text{Fil}^f(\mathcal{A})) $$ is $\text{FAc}(\mathcal{A})$ and the functor $H^0 \circ \text{gr}$ factors through $Q$.

Proof. We know that $H^0 \circ \text{gr}$ is a homological functor by Lemma 13.14.3. Thus this lemma is a special case of Lemma 13.6.11. $\square$

Definition 13.14.5. Let $\mathcal{A}$ be an abelian category. Let $\text{FAc}(\mathcal{A})$ and $\text{FQis}(\mathcal{A})$ be as in Lemma 13.14.4. 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 13.14.6. The functors $\text{gr}^p, \text{gr}, (\text{forget }F)$ induce canonical exact functors $$ \text{gr}^p, \text{gr}, (\text{forget }F): DF(\mathcal{A}) \longrightarrow D(\mathcal{A}) $$ which commute with the localization functors.

Proof. This follows from the universal property of localization, see Lemma 13.5.6, 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 13.14.4. 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 12.16.15. $\square$

Definition 13.14.7. 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 13.14.8. Let $\mathcal{A}$ be an abelian category. Let $K^\bullet \in K(\text{Fil}^f(\mathcal{A}))$.

  1. 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 $n < a$.
  2. 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 $n > b$.
  3. If $H^n(\text{gr}(K^\bullet)) = 0$ for all $|n| \gg 0$, then there exists a commutative diagram of morphisms of complexes $$ \xymatrix{ K^\bullet \ar[r] & L^\bullet \\ M^\bullet \ar[u] \ar[r] & N^\bullet \ar[u] } $$ where all the arrows are filtered quasi-isomorphisms, $L^\bullet$ bounded below, $M^\bullet$ bounded above, and $N^\bullet$ a bounded complex.

Proof. Suppose that $H^n(\text{gr}(K^\bullet)) = 0$ for all $n < a$. By Homology, Lemma 12.16.15 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 12.16.13. 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 13.14.9. 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. $\text{FQis}^b(\mathcal{A})$.

  1. 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 $$ K^{+}(\text{Fil}^f\mathcal{A})/\text{FAc}^{+}(\mathcal{A}) = \text{FQis}^{+}(\mathcal{A})^{-1}K^{+}(\text{Fil}^f\mathcal{A}) \longrightarrow DF^{+}(\mathcal{A}) $$
  2. 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 $$ K^{-}(\text{Fil}^f\mathcal{A})/\text{FAc}^{-}(\mathcal{A}) = \text{FQis}^{-}(\mathcal{A})^{-1}K^{-}(\text{Fil}^f\mathcal{A}) \longrightarrow DF^{-}(\mathcal{A}) $$
  3. 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 $$ K^b(\text{Fil}^f\mathcal{A})/\text{FAc}^b(\mathcal{A}) = \text{FQis}^b(\mathcal{A})^{-1}K^b(\text{Fil}^f\mathcal{A}) \longrightarrow DF^b(\mathcal{A}) $$

Proof. This follows from the results above, in particular Lemma 13.14.8, by exactly the same arguments as used in the proof of Lemma 13.11.6. $\square$

    The code snippet corresponding to this tag is a part of the file derived.tex and is located in lines 4017–4351 (see updates for more information).

    \section{Filtered derived categories}
    \label{section-filtered-derived-category}
    
    \noindent
    A reference for this section is \cite[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 \cite{Buhler}, \cite{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 \ref{lemma-two-ss-complex-functor}, see also
    Remark \ref{remark-functorial-ss}.
    
    \medskip\noindent
    We will use the notation regarding filtered objects introduced in
    Homology, Section \ref{homology-section-filtrations}.
    The category of filtered objects of $\mathcal{A}$ is
    denoted $\text{Fil}(\mathcal{A})$.
    All filtrations will be decreasing by fiat.
    
    \begin{definition}
    \label{definition-finite-filtered}
    Let $\mathcal{A}$ be an abelian category. The
    {\it 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 $\text{Fil}^f(\mathcal{A})$.
    \end{definition}
    
    \noindent
    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 \ref{homology-definition-graded}.
    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 \ref{homology-example-not-abelian}.
    
    \medskip\noindent
    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 \ref{lemma-additive-exact-homotopy-category}.
    By analogy with the case of the homotopy category of an abelian category
    we make the following definitions.
    
    \begin{definition}
    \label{definition-filtered-acyclic}
    Let $\mathcal{A}$ be an abelian category.
    \begin{enumerate}
    \item Let $\alpha : K^\bullet \to L^\bullet$ be a morphism of
    $K(\text{Fil}^f(\mathcal{A}))$. We say that
    $\alpha$ is a {\it filtered quasi-isomorphism} if
    the morphism $\text{gr}(\alpha)$ is a quasi-isomorphism.
    \item Let $K^\bullet$ be an object of $K(\text{Fil}^f(\mathcal{A}))$.
    We say that $K^\bullet$ is {\it filtered acyclic} if
    the complex $\text{gr}(K^\bullet)$ is acyclic.
    \end{enumerate}
    \end{definition}
    
    \noindent
    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.
    
    \begin{lemma}
    \label{lemma-filtered-cohomology-homological}
    Let $\mathcal{A}$ be an abelian category.
    \begin{enumerate}
    \item The functor
    $K(\text{Fil}^f(\mathcal{A})) \longrightarrow \text{Gr}(\mathcal{A})$,
    $K^\bullet \longmapsto H^0(\text{gr}(K^\bullet))$
    is homological.
    \item The functor
    $K(\text{Fil}^f(\mathcal{A})) \rightarrow \mathcal{A}$,
    $K^\bullet \longmapsto H^0(\text{gr}^p(K^\bullet))$
    is homological.
    \item The functor
    $K(\text{Fil}^f(\mathcal{A})) \longrightarrow \mathcal{A}$,
    $K^\bullet \longmapsto H^0((\text{forget }F)K^\bullet)$
    is homological.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    This follows from the fact that
    $H^0 : K(\mathcal{A}) \to \mathcal{A}$ is homological, see
    Lemma \ref{lemma-cohomology-homological}
    and the fact that the functors $\text{gr}, \text{gr}^p, (\text{forget }F)$
    are exact functors of triangulated categories. See
    Lemma \ref{lemma-exact-compose-homological-functor}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-filtered-acyclic}
    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 \ref{lemma-operations})
    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
    $$
    Q :
    K(\text{Fil}^f(\mathcal{A}))
    \longrightarrow
    \text{FQis}(\mathcal{A})^{-1}K(\text{Fil}^f(\mathcal{A}))
    $$
    is $\text{FAc}(\mathcal{A})$ and the functor $H^0 \circ \text{gr}$
    factors through $Q$.
    \end{lemma}
    
    \begin{proof}
    We know that $H^0 \circ \text{gr}$ is a homological functor by
    Lemma \ref{lemma-filtered-cohomology-homological}.
    Thus this lemma is a special case of
    Lemma \ref{lemma-acyclic-general}.
    \end{proof}
    
    \begin{definition}
    \label{definition-filtered-derived}
    Let $\mathcal{A}$ be an abelian category.
    Let $\text{FAc}(\mathcal{A})$ and $\text{FQis}(\mathcal{A})$
    be as in
    Lemma \ref{lemma-filtered-acyclic}.
    The {\it filtered derived category of $\mathcal{A}$}
    is the triangulated category
    $$
    DF(\mathcal{A}) =
    K(\text{Fil}^f(\mathcal{A}))/\text{FAc}(\mathcal{A}) =
    \text{FQis}(\mathcal{A})^{-1} K(\text{Fil}^f(\mathcal{A})).
    $$
    \end{definition}
    
    \begin{lemma}
    \label{lemma-filtered-derived-functors}
    The functors $\text{gr}^p, \text{gr}, (\text{forget }F)$ induce
    canonical exact functors
    $$
    \text{gr}^p, \text{gr}, (\text{forget }F):
    DF(\mathcal{A})
    \longrightarrow
    D(\mathcal{A})
    $$
    which commute with the localization functors.
    \end{lemma}
    
    \begin{proof}
    This follows from the universal property of localization, see
    Lemma \ref{lemma-universal-property-localization},
    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 \ref{lemma-filtered-acyclic}.
    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 \ref{homology-lemma-filtered-acyclic}.
    \end{proof}
    
    \begin{definition}
    \label{definition-filtered-derived-bounded}
    Let $\mathcal{A}$ be an abelian category.
    The {\it 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
    $DF^{-}(\mathcal{A})$.
    \end{definition}
    
    \begin{lemma}
    \label{lemma-filtered-complex-cohomology-bounded}
    Let $\mathcal{A}$ be an abelian category.
    Let $K^\bullet \in K(\text{Fil}^f(\mathcal{A}))$.
    \begin{enumerate}
    \item 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 $n < a$.
    \item 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 $n > b$.
    \item If $H^n(\text{gr}(K^\bullet)) = 0$ for all $|n| \gg 0$, then there
    exists a commutative diagram of morphisms of complexes
    $$
    \xymatrix{
    K^\bullet \ar[r] & L^\bullet \\
    M^\bullet \ar[u] \ar[r] & N^\bullet \ar[u]
    }
    $$
    where all the arrows are filtered quasi-isomorphisms, $L^\bullet$
    bounded below, $M^\bullet$ bounded above, and $N^\bullet$ a bounded
    complex.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Suppose that $H^n(\text{gr}(K^\bullet)) = 0$ for all $n < a$. By
    Homology, Lemma \ref{homology-lemma-filtered-acyclic}
    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
    $\Coim(d^{a - 1}) = \Im(d^{a - 1})$ in $\text{Fil}^f(\mathcal{A})$
    and the map $\text{gr}(\Im(d^{a - 1})) \to \text{gr}(K^a)$
    is injective with image equal to the image of
    $\text{gr}(K^{a - 1}) \to \text{gr}(K^a)$, see
    Homology, Lemma \ref{homology-lemma-characterize-strict}.
    This means that the map $K^\bullet \to \tau_{\geq a}K^\bullet$
    into the truncation
    $$
    \tau_{\geq a}K^\bullet =
    (\ldots \to 0 \to K^a/\Im(d^{a - 1}) \to K^{a + 1} \to \ldots)
    $$
    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).
    \end{proof}
    
    \noindent
    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 $\text{FQis}(\mathcal{A})$.
    
    \begin{lemma}
    \label{lemma-filtered-bounded-derived}
    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 \ref{lemma-operations})
    are the sets $\text{FQis}^{+}(\mathcal{A})$, $\text{FQis}^{-}(\mathcal{A})$,
    resp.\ $\text{FQis}^b(\mathcal{A})$.
    \begin{enumerate}
    \item 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
    $$
    K^{+}(\text{Fil}^f\mathcal{A})/\text{FAc}^{+}(\mathcal{A}) =
    \text{FQis}^{+}(\mathcal{A})^{-1}K^{+}(\text{Fil}^f\mathcal{A})
    \longrightarrow
    DF^{+}(\mathcal{A})
    $$
    \item 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
    $$
    K^{-}(\text{Fil}^f\mathcal{A})/\text{FAc}^{-}(\mathcal{A}) =
    \text{FQis}^{-}(\mathcal{A})^{-1}K^{-}(\text{Fil}^f\mathcal{A})
    \longrightarrow
    DF^{-}(\mathcal{A})
    $$
    \item 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
    $$
    K^b(\text{Fil}^f\mathcal{A})/\text{FAc}^b(\mathcal{A}) =
    \text{FQis}^b(\mathcal{A})^{-1}K^b(\text{Fil}^f\mathcal{A})
    \longrightarrow
    DF^b(\mathcal{A})
    $$
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    This follows from the results above, in particular
    Lemma \ref{lemma-filtered-complex-cohomology-bounded},
    by exactly the same arguments as used in the proof of
    Lemma \ref{lemma-bounded-derived}.
    \end{proof}

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