Filtered derived categories

Written by Claude Opus 5.5 (Anthropic) on 22 October 2026. Self-checked by the writing AI. Public domain (CC0).

A short exact sequence of complexes \(0\to L'\to L\to L''\to0\) determines a distinguished triangle, but a morphism of triangles need not come from a morphism of short exact sequences, and some invariants of an endomorphism, such as its trace, are not determined by the triangle alone. A finite filtration \(F^pL\) of a complex records such a flag. The filtered derived category keeps the flag while inverting the maps that are quasi-isomorphisms on every graded piece \(\operatorname{gr}^pL=F^pL/F^{p+1}L\). The lesson proves:

Prerequisites. From the AI-integrated Stacks project: filtered objects, strict morphisms and their graded pieces [Stacks, Tag 0120], [Tag 0123], [Tag 0127]; complexes of filtered objects with exact graded pieces [Tag 05QH]; pushouts of filtered objects [Tag 05SM]; truncations [Tag 0118]; the spectral sequence of a filtered complex [Tag 012M], its first differential [Tag 012N] and its convergence [Tag 012W]; localization and its morphisms [Tag 04VB], [Tag 05Q4]; essentially constant systems [Tag 05PU]; the homotopy category of an additive category and its distinguished triangles [Tag 014S], [Tag 014Q], [Tag 014M], [Tag 014X]; triangulated subcategories [Tag 05QX]; Verdier quotients [Tag 05R6], [Tag 05RJ], [Tag 05RM]; homological functors [Tag 05RS], [Tag 05QZ], [Tag 0149]; the triangle of a short exact sequence of complexes [Tag 0152]; injective resolutions [Tag 013K], maps from acyclic complexes into bounded below complexes of injectives [Tag 013R], [Tag 05TG], [Tag 05TH]; right derived functors [Tag 05S9]; Ext groups and their composition [Tag 06XP], Yoneda extensions in an abelian category [Tag 06XU]; filtered injective objects [Tag 05TP]; the filtered derived category of the Stacks project [Tag 05RX], [Tag 015Q].

Conventions

\(\mathcal A\) is an abelian category. Complexes have differentials of degree \(+1\). For a complex \(X\), \(X[n]^i=X^{n+i}\) with differential \((-1)^nd_X\); a morphism \(f\) is shifted without sign, \(f[n]^i=f^{n+i}\). The cone of \(f\colon X\to Y\) is \(C(f)^n=Y^n\oplus X^{n+1}\) with \(d(y,x)=(dy+f(x),-dx)\), and the triangles \((X,Y,C(f),f,i,-p)\) and the triangles isomorphic to them are the distinguished triangles of the homotopy category [Tag 014Q]. A short exact sequence of complexes \(0\to X\to Y\to Z\to0\) defines the distinguished triangle \(X\to Y\to Z\xrightarrow{\delta}X[1]\) of [Tag 0152], with \(\delta=-p\circ q^{-1}\) for \(q\colon C(X\to Y)\to Z\) and \(p\colon C(X\to Y)\to X[1]\) the obvious maps; \(H^n(\delta)\) is the connecting map of the long exact cohomology sequence (lift a cycle of \(Z\), apply \(d\), and pull back to \(X\)). The Hom complex of two complexes is \(\operatorname{Hom}^n(X,Y)=\prod_k\operatorname{Hom}(X^k,Y^{k+n})\) with \(\mathrm d f=d_Y\circ f-(-1)^nf\circ d_X\); its \(n\)-cycles are the morphisms of complexes \(X\to Y[n]\) and \(H^n\operatorname{Hom}^\bullet(X,Y)=\operatorname{Hom}_{K(\mathcal A)}(X,Y[n])\). Truncations are \(\tau_{\leq n}X=\) \((\cdots\to X^{n-1}\to\operatorname{Ker}d^n\to0\to\cdots)\) and \(\tau_{\geq n}X=\) \((\cdots\to0\to\operatorname{Coker}d^{n-1}\to X^{n+1}\to\cdots)\) [Tag 0118]. For \(\xi\in\operatorname{Ext}^i(X,Y)=\operatorname{Hom}_{D(\mathcal A)}(X,Y[i])\) and \(\eta\in\operatorname{Ext}^j(Y,Z)\), \(\eta\circ\xi=\eta[i]\circ\xi\in\operatorname{Ext}^{i+j}(X,Z)\) [Tag 06XP]. Localizations may have morphism classes that are not sets; as in the Stacks project this causes no difficulty, and when \(\mathcal A\) has enough injectives the morphism classes of the categories below are sets (Proposition 4.6).

1. Filtered and graded complexes

Definition 1.1. A filtered object of \(\mathcal A\) is an object \(X\) with subobjects \(F^{p+1}X\subset F^pX\), \(p\in\mathbf Z\); it has filtration in \([a,b]\) (integers \(a\le b\)) if \(F^aX=X\) and \(F^{b+1}X=0\), and finite filtration if this holds for some \(a\le b\). Morphisms satisfy \(f(F^pX)\subset F^pY\). They form the additive categories \(\operatorname{Fil}(\mathcal A)\supset\operatorname{Fil}_f(\mathcal A)\) [Tag 0120]; \(\operatorname{gr}^pX=F^pX/F^{p+1}X\).

A filtered complex is a complex \(L\) of \(\mathcal A\) with subcomplexes \(F^{p+1}L\subset F^pL\) such that \(F^aL=L\) and \(F^{b+1}L=0\) for some integers \(a\le b\), the same in all degrees ("filtration in \([a,b]\)"). Morphisms are morphisms of complexes with \(f(F^pL)\subset F^pM\). The category is \(CF(\mathcal A)\), with full subcategories \(CF^{[a,b]}(\mathcal A)\); it is the full subcategory of the category of complexes of \(\operatorname{Fil}(\mathcal A)\) formed by the complexes whose filtration is finite uniformly in the degree. Put \(\operatorname{gr}^pL=F^pL/F^{p+1}L\) and \(F_pL=L/F^pL\); the epimorphisms \(F_{p+1}L\to F_pL\) form the cofiltration.

A graded complex is a complex \(E\) with a decomposition \(E=\bigoplus_pk^pE\) into subcomplexes, only finitely many of them nonzero; morphisms preserve the decompositions. The category is \(CG(\mathcal A)\), with full subcategories \(CG^{[a,b]}(\mathcal A)\) (\(k^pE=0\) for \(p\notin[a,b]\)). The associated graded is the additive functor \(\operatorname{gr}\colon CF(\mathcal A)\to CG(\mathcal A)\), \(k^p\operatorname{gr}L=\operatorname{gr}^pL\).

The twist \(L(n)\) has the same underlying complex and \(F^p(L(n))=F^{p+n}L\); for graded complexes \(k^p(E(n))=k^{p+n}E\). The shift \(L[n]\) carries \(F^p(L[n])=(F^pL)[n]\). The functor \(\operatorname{gr}\) commutes with shifts and twists.

Truncations. For \(L\in CF(\mathcal A)\) and \(n\in\mathbf Z\), \(\tau_{\leq n}L\) is filtered by the subcomplexes \(\tau_{\leq n}(F^pL)\) (the truncation preserves monomorphisms), and \(\tau_{\geq n}L\) by \(F^p(\tau_{\geq n}L)=\operatorname{Ker}(\tau_{\geq n}L\to\tau_{\geq n}(F_pL))\) (the truncation preserves epimorphisms, so that \(F_p(\tau_{\geq n}L)=\tau_{\geq n}(F_pL)\)). Concretely, \(\tau_{\leq n}L\) carries in degree \(n\) the filtration of \(\operatorname{Ker}d^n\) induced from \(L^n\), and \(\tau_{\geq n}L\) carries in degree \(n\) the image filtration of \(\operatorname{Coker}d^{n-1}\); in the other nonzero degrees the filtration is that of \(L\). Both have filtration in \([a,b]\) when \(L\) has, and the canonical maps \(\tau_{\leq n}L\to L\to\tau_{\geq n}L\) are morphisms of \(CF(\mathcal A)\). For \(M\in CF(\mathcal A)\) vanishing in degrees \(<n\) (resp. \(N\) vanishing in degrees \(>n\)), every morphism \(L\to M\) (resp. \(N\to L\)) factors uniquely through \(L\to\tau_{\geq n}L\) (resp. \(\tau_{\leq n}L\to L\)), since this holds for the underlying complexes and the factorization is filtered. Graded complexes are truncated componentwise. The functor \(\operatorname{gr}\) does not commute with these truncations (Exercise 6.2).

Cones. For \(f\colon L\to M\) in \(CF(\mathcal A)\), the cone \(C(f)\) with \(F^pC(f)=F^pM\oplus F^pL[1]\) lies in \(CF(\mathcal A)\), the sequence \(0\to M\to C(f)\to L[1]\to0\) is split in each degree as a sequence of filtered objects, and \(\operatorname{gr}C(f)=C(\operatorname{gr}f)\). The same holds in \(CG(\mathcal A)\).

Lemma 1.2. Let \(L\in CF(\mathcal A)\) and \(n,p\in\mathbf Z\).

  1. \(H^q(F_pL)\to H^q(F_p\tau_{\geq n}L)\) is an isomorphism for \(q\ge n\), and \(H^q(F^pL)\to H^q(F^p\tau_{\geq n}L)\) is an isomorphism for \(q\ge n+1\).
  2. \(H^q(F^p\tau_{\leq n}L)\to H^q(F^pL)\) is an isomorphism for \(q\le n\), and \(H^q(F_p\tau_{\leq n}L)\to H^q(F_pL)\) is an isomorphism for \(q\le n-1\).

Proof. Since \(F_p\tau_{\geq n}L=\tau_{\geq n}F_pL\) and \(F^p\tau_{\leq n}L=\tau_{\leq n}F^pL\), the first halves are the corresponding properties of truncations of complexes. For the second half of (1) compare the long exact sequences of \(0\to F^pX\to X\to F_pX\to0\) for \(X=L\) and \(X=\tau_{\geq n}L\) (exact by definition of the filtration of \(\tau_{\geq n}L\)): in \[ \begin{aligned}&H^{q-1}(X) \to H^{q-1}(F_pX)\\&\quad \to H^q(F^pX) \to H^q(X)\\&\quad \to H^q(F_pX),\end{aligned} \] the maps induced on the four outer terms are isomorphisms when \(q-1\ge n\), so the five lemma gives the claim. For (2) use \(0\to F^p\tau_{\leq n}L\to\tau_{\leq n}L\to F_p\tau_{\leq n}L\to0\) and the segment \(H^q(F^pX)\to\) \(H^q(X)\to\) \(H^q(F_pX)\to\) \(H^{q+1}(F^pX)\to\) \(H^{q+1}(X)\), whose outer maps are isomorphisms when \(q+1\le n\). \(\square\)

In the excluded degrees \(q=n\) (for \(F^p\tau_{\geq n}\)) and \(q=n\) (for \(F_p\tau_{\leq n}\)) the maps need not be isomorphisms (Exercises 6.1 and 6.2).

2. The filtered derived category

Lemma 2.1. For a morphism \(f\colon L\to M\) of \(CF(\mathcal A)\) the following are equivalent: (i) \(F_pf\) is a quasi-isomorphism for every \(p\); (ii) \(F^pf\) is a quasi-isomorphism for every \(p\); (iii) \(\operatorname{gr}^pf\) is a quasi-isomorphism for every \(p\); (iv) \(\operatorname{gr}C(f)\) is acyclic.

Proof. Let both filtrations be in \([a,b]\). (iii)\(\Rightarrow\)(ii) by descending induction on \(p\), starting from \(F^{b+1}=0\), with the five lemma on \(0\to F^{p+1}\to F^p\to\operatorname{gr}^p\to0\); (ii)\(\Rightarrow\)(iii) by the five lemma on the same sequences. (i)\(\Leftrightarrow\)(ii): \(f=F^af=F_{b+1}f\), and the five lemma on \(0\to F^pL\to L\to F_pL\to0\) gives each implication. (iii)\(\Leftrightarrow\)(iv) because \(\operatorname{gr}C(f)=C(\operatorname{gr}f)\) and a morphism is a quasi-isomorphism exactly when its cone is acyclic. \(\square\)

Such an \(f\) is a filtered quasi-isomorphism; \(L\) is filtered acyclic if \(\operatorname{gr}L\) is acyclic, equivalently if every \(F^pL\), or every \(F_pL\), is acyclic. A morphism of \(CG(\mathcal A)\) is a graded quasi-isomorphism if each component is a quasi-isomorphism.

A filtered homotopy between \(f,g\colon L\to M\) is a homotopy \(h\) with \(h(F^pL^n)\subset F^pM^{n-1}\); \(KF(\mathcal A)\) has the objects of \(CF(\mathcal A)\) and filtered homotopy classes of morphisms. Filtered homotopic morphisms induce the same maps on \(H(F^p-)\), \(H(F_p-)\) and \(H(\operatorname{gr}^p-)\), so filtered quasi-isomorphisms are defined in \(KF(\mathcal A)\). Graded homotopies and \(KG(\mathcal A)\) are defined in the same way.

Proposition 2.2.

  1. \(KF(\mathcal A)\) is a full triangulated subcategory of the homotopy category \(K(\operatorname{Fil}(\mathcal A))\); its distinguished triangles are the triangles isomorphic to \((L,M,C(f),f,i,-p)\).
  2. The filtered acyclic complexes form a strictly full saturated triangulated subcategory \(FAc(\mathcal A)\) of \(KF(\mathcal A)\); the corresponding saturated multiplicative system is the set \(FQis(\mathcal A)\) of filtered quasi-isomorphisms.
  3. The filtered derived category \(DF(\mathcal A)=\) \(KF(\mathcal A)/FAc(\mathcal A)=\) \(FQis(\mathcal A)^{-1}KF(\mathcal A)\) is triangulated and the quotient functor \(Q\) is exact; its morphisms are given by (2.1) below.
  4. \(F^p\), \(F_p\), \(\operatorname{gr}^p\) and the underlying complex induce exact functors \(DF(\mathcal A)\to D(\mathcal A)\), and \(\operatorname{gr}\) induces an exact functor \(DF(\mathcal A)\to DG(\mathcal A)\), where \(DG(\mathcal A)=KG(\mathcal A)/GAc(\mathcal A)\) is defined in the same way from the graded acyclic complexes. A morphism \(\varphi\) of \(DF(\mathcal A)\) is an isomorphism if and only if \(\operatorname{gr}^p\varphi\) is an isomorphism of \(D(\mathcal A)\) for every \(p\).

In (3), for \(L,M\in KF(\mathcal A)\), \[ \begin{gathered}\operatorname{Hom}_{DF}(L,M)\\ =\operatorname*{colim}_{s\colon L'\to L}\operatorname{Hom}_{KF}(L',M)\\ =\operatorname*{colim}_{t\colon M\to M'}\operatorname{Hom}_{KF}(L,M'),\end{gathered} \tag{2.1} \] the colimits running over filtered quasi-isomorphisms.

Proof. (1) \(\operatorname{Fil}(\mathcal A)\) is additive, so \(K(\operatorname{Fil}(\mathcal A))\) is triangulated with the cone triangles [Tag 014S]. Filtered homotopies are the homotopies of complexes of \(\operatorname{Fil}(\mathcal A)\), so \(KF(\mathcal A)\) is a full subcategory; it is stable under shifts and cones, hence a triangulated subcategory [Tag 05QX]. (2) Each \(\operatorname{gr}^p\colon\operatorname{Fil}(\mathcal A)\to\mathcal A\) is additive and induces an exact functor \(K(\operatorname{Fil}(\mathcal A))\to K(\mathcal A)\) [Tag 014X]; with \(H^0\) homological [Tag 05RS] this gives a homological functor \(H=(H^0\operatorname{gr}^p)_{p\in\mathbf Z}\colon KF(\mathcal A)\to\mathcal A^{\mathbf Z}\) [Tag 05QZ], into the abelian category of \(\mathbf Z\)-indexed families. Its kernel, the objects with \(H(X[i])=0\) for all \(i\), is \(FAc(\mathcal A)\), and the morphisms inverted by all \(H(-[i])\) are the filtered quasi-isomorphisms; [Tag 05RM] gives (2). (3) is [Tag 05R6] with the description of morphisms in a localization [Tag 05Q4] (and its dual). (4) These functors are exact on \(KF(\mathcal A)\) and send filtered quasi-isomorphisms to quasi-isomorphisms (Lemma 2.1), so they factor through \(DF(\mathcal A)\) [Tag 05RJ]. If every \(\operatorname{gr}^p\varphi\) is an isomorphism, complete \(\varphi\) to a distinguished triangle \(X\to Y\to Z\to X[1]\) of \(DF(\mathcal A)\); the long exact sequences of the homological functors \(H^q\operatorname{gr}^p\) give \(H^q(\operatorname{gr}^pZ)=0\) for all \(p,q\), so \(Z\) lies in \(FAc(\mathcal A)\) and \(Z\cong0\); then \(\operatorname{Hom}(W,\varphi)\) is bijective for every \(W\) [Tag 0149], and \(\varphi\) is an isomorphism. \(\square\)

Proposition 2.3. Let \(\operatorname{Fam}(D(\mathcal A))\) be the category of families \((X_p)_{p\in\mathbf Z}\) of objects of \(D(\mathcal A)\), only finitely many nonzero, with \(\operatorname{Hom}((X_p),(Y_p))=\) \(\prod_p\operatorname{Hom}_{D(\mathcal A)}(X_p,Y_p)\), triangulated componentwise. The functor \(DG(\mathcal A)\to\operatorname{Fam}(D(\mathcal A))\), \(E\mapsto(k^pE)_p\), is an equivalence of triangulated categories.

Proof. The functor is defined and exact because \(KG(\mathcal A)\to\operatorname{Fam}(D(\mathcal A))\) is exact and kills graded acyclic complexes. It is essentially surjective. Let \(E,F\) be supported in \([a,b]\). In (2.1) for \(DG\), a graded quasi-isomorphism \(s\colon E'\to E\) can be replaced by its restriction to \(\bigoplus_{p\in[a,b]}k^pE'\), still a graded quasi-isomorphism because \(k^pE'\) is acyclic for \(p\notin[a,b]\). The category of graded quasi-isomorphisms \(E'\to E\) with \(E'\) supported in \([a,b]\) is the product over \(p\in[a,b]\) of the categories of quasi-isomorphisms \(X\to k^pE\), and \(\operatorname{Hom}_{KG}(E',F)=\) \(\prod_{p\in[a,b]}\operatorname{Hom}_K(k^pE',k^pF)\). A colimit over a finite product of filtered categories of a product of functors each depending on one factor is the product of the colimits, so \(\operatorname{Hom}_{DG}(E,F)=\) \(\prod_p\operatorname{Hom}_{D(\mathcal A)}(k^pE,k^pF)\). \(\square\)

Constructions on \(DG(\mathcal A)\) therefore reduce to componentwise constructions in \(D(\mathcal A)\); we keep the graded language because the grading organizes the bookkeeping.

Definition 2.4. For \(L,M\in DF(\mathcal A)\) and \(n\in\mathbf Z\), \(\operatorname{Ext}^n_{\mathrm{fil}}(L,M)=\operatorname{Hom}_{DF}(L,M[n])\); for \(E,F\in DG(\mathcal A)\), \(\operatorname{Ext}^n_{\mathrm{grad}}(E,F)=\) \(\operatorname{Hom}_{DG}(E,F[n])=\) \(\prod_p\operatorname{Ext}^n(k^pE,k^pF)\).

A distinguished triangle in either variable gives long exact sequences of these groups [Tag 0149], and since \(\operatorname{gr}\) is exact, \(f\mapsto\operatorname{gr}f\) is a morphism from the filtered to the graded long exact sequences.

3. Bounded objects and filtration intervals

Lemma 3.1. Let \(L\in CF(\mathcal A)\) and \(n\in\mathbf Z\). The following are equivalent: (i) \(H^q(\operatorname{gr}^pL)=0\) for \(q<n\) and all \(p\); (ii) \(H^q(F^pL)=H^q(F_pL)=0\) for \(q<n\) and all \(p\); (iii) \(L\to\tau_{\geq n}L\) is a filtered quasi-isomorphism. Likewise (i') \(H^q(\operatorname{gr}^pL)=0\) for \(q>n\), (ii') \(H^q(F^pL)=H^q(F_pL)=0\) for \(q>n\), and (iii') \(\tau_{\leq n}L\to L\) is a filtered quasi-isomorphism are equivalent.

Proof. (i)\(\Rightarrow\)(ii): from \(F^{b+1}L=0\) and \(0\to F^{p+1}L\to F^pL\to\operatorname{gr}^pL\to0\), descending induction gives \(H^q(F^pL)=0\) for \(q<n\); from \(F_aL=0\) and \(0\to\operatorname{gr}^pL\to F_{p+1}L\to F_pL\to0\), ascending induction gives \(H^q(F_pL)=0\) for \(q<n\). (ii)\(\Rightarrow\)(iii): by Lemma 1.2 (1), \(H^q(F_pL)\to H^q(F_p\tau_{\geq n}L)\) is an isomorphism for \(q\ge n\), and for \(q<n\) both groups vanish; so \(F_p\) of the map is a quasi-isomorphism for every \(p\), and Lemma 2.1 applies. (iii)\(\Rightarrow\)(i): \(\operatorname{gr}L\) is quasi-isomorphic to \(\operatorname{gr}\tau_{\geq n}L\), which vanishes in degrees \(<n\). The second set of equivalences is proved in the same way with Lemma 1.2 (2). \(\square\)

Definition 3.2. \(D^{\geq n}F(\mathcal A)\) (resp. \(D^{\leq n}F(\mathcal A)\)) is the full subcategory of \(DF(\mathcal A)\) of objects with \(H^q(\operatorname{gr}^pL)=0\) for \(q<n\) (resp. \(q>n\)) and all \(p\). For an interval \([a,b]\) of \(\mathbf Z\cup\{\pm\infty\}\), \(D^{[a,b]}F=D^{\geq a}F\cap D^{\leq b}F\), with \(D^{\geq-\infty}F=D^{\leq+\infty}F=DF\); \(D^+F=\bigcup_nD^{\geq n}F\), \(D^-F=\bigcup_nD^{\leq n}F\), \(D^bF=D^+F\cap D^-F\). The same notation is used for \(DG(\mathcal A)\) with the cohomology of the components. These conditions depend only on the isomorphism class in \(DF(\mathcal A)\); \(D^+F\), \(D^-F\) and \(D^bF\) are triangulated subcategories. \(C^{[a,b]}F(\mathcal A)\) denotes the full subcategory of \(CF(\mathcal A)\) of complexes with \(L^q=0\) for \(q\notin[a,b]\).

Proposition 3.3. For every interval \([a,b]\) of \(\mathbf Z\cup\{\pm\infty\}\), the functor \(C^{[a,b]}F(\mathcal A)\to DF(\mathcal A)\) induces an equivalence \[ C^{[a,b]}F(\mathcal A)[FQis^{-1}]\longrightarrow D^{[a,b]}F(\mathcal A), \] where the source is the localization [Tag 04VB] of the category of filtered complexes with terms in degrees \([a,b]\) and filtered morphisms of complexes at the filtered quasi-isomorphisms between them. In particular every object of \(D^{[a,b]}F(\mathcal A)\) is isomorphic to a complex with terms in degrees \([a,b]\). The same holds for graded complexes.

Proof. Let \(CF_{[a,b]}\subset CF(\mathcal A)\) and \(KF_{[a,b]}\subset KF(\mathcal A)\) be the full subcategories of complexes with \(H^q(\operatorname{gr}L)=0\) for \(q\notin[a,b]\), and \(W\) the filtered quasi-isomorphisms between their objects.

(a) \(KF_{[a,b]}[W^{-1}]\to DF(\mathcal A)\) is fully faithful with essential image \(D^{[a,b]}F(\mathcal A)\). The axioms of a multiplicative system hold for \(W\) in \(KF_{[a,b]}\): the objects that the axioms for \(FQis\) in \(KF(\mathcal A)\) produce are isomorphic in \(DF(\mathcal A)\) to objects of \(KF_{[a,b]}\), hence lie in \(KF_{[a,b]}\), the defining condition depending only on the isomorphism class. For \(L\in KF_{[a,b]}\), the source of every filtered quasi-isomorphism \(s\colon L'\to L\) lies in \(KF_{[a,b]}\). So the colimits computing morphisms in the two localizations [Tag 05Q4] run over the same categories.

(b) \(CF_{[a,b]}[W^{-1}]\to KF_{[a,b]}[W^{-1}]\) is an isomorphism of categories. It suffices that filtered homotopic morphisms \(f,g\colon L\to M\) become equal in \(CF_{[a,b]}[W^{-1}]\). Let \(\operatorname{Cyl}(L)^n=L^n\oplus L^{n+1}\oplus L^n\) with \(d(x,y,z)=(dx+y,-dy,dz-y)\), filtered termwise, with \(i_0(x)=(x,0,0)\), \(i_1(z)=(0,0,z)\), \(\pi(x,y,z)=x+z\). These are morphisms of \(CF(\mathcal A)\) and \(\pi i_0=\pi i_1=\mathrm{id}\). The map \(\pi\) is surjective with kernel \(\{(x,y,-x)\}\), which is isomorphic as a filtered complex to the cone of \(\mathrm{id}_L\), filtered acyclic; so \(\pi\in W\) and \(\operatorname{Cyl}(L)\in CF_{[a,b]}\). For a filtered homotopy \(h\) with \(f-g=dh+hd\), \(H(x,y,z)=f(x)+h(y)+g(z)\) is a morphism of filtered complexes with \(Hi_0=f\), \(Hi_1=g\). In the localization \(\pi\) is invertible, so \(i_0=i_1\) and \(f=g\).

(c) The inclusion \(\iota\colon C^{[a,b]}F(\mathcal A)\to CF_{[a,b]}\) and \(\tau=\tau_{\geq a}\tau_{\leq b}\colon CF_{[a,b]}\to C^{[a,b]}F(\mathcal A)\) (omit a truncation at an infinite end) preserve filtered quasi-isomorphisms (Lemma 3.1), \(\tau\iota=\mathrm{id}\), and \(X\leftarrow\tau_{\leq b}X\to\iota\tau X\) is a natural zigzag of filtered quasi-isomorphisms (Lemma 3.1). Hence \(\iota\) and \(\tau\) induce inverse equivalences of the localizations at \(W\). Combining (a), (b) and (c) proves the proposition; the graded case is the same, componentwise. \(\square\)

Lemma 3.4. Let \(L'\) be a complex of \(\operatorname{Fil}(\mathcal A)\) whose terms have finite filtrations (not necessarily uniformly in the degree), and let \(a\le b\) be integers such that \(\operatorname{gr}^pL'\) is acyclic for \(p\notin[a,b]\). Then \(F^{b+1}L'\) and \(L'/F^aL'\) are acyclic. Let \(L''\) be the complex \(F^aL'\) with the filtration \(F^pL''=F^aL'\) for \(p\le a\), \(F^pL''=F^pL'\) for \(a\le p\le b\), \(F^pL''=0\) for \(p>b\). Then \(L''\) has filtration in \([a,b]\), and the inclusion \(L''\to L'\) is a filtered morphism inducing quasi-isomorphisms on all graded pieces.

Proof. Fix \(n\). There is \(N\) with \(F^NL'\) zero in degrees \(n-1,n,n+1\), so \(H^n(F^NL')=0\) and \(H^n(F^{b+1}L')\to H^n(F^{b+1}L'/F^NL')\) is injective; the target vanishes, because \(F^{b+1}L'/F^NL'\) has a finite filtration with acyclic graded pieces \(\operatorname{gr}^pL'\), \(b<p<N\). So \(F^{b+1}L'\) is acyclic. Dually, there is \(N'\) with \(F^{N'}L'=L'\) in degrees \(n-1,n,n+1\), so \(H^n(F^{N'}L'/F^aL')\to H^n(L'/F^aL')\) is surjective with acyclic source; hence \(L'/F^aL'\) is acyclic. The inclusion \(L''\to L'\) induces the identity on \(\operatorname{gr}^p\) for \(a\le p<b\), the projection \(F^bL'\to\operatorname{gr}^bL'\) on \(\operatorname{gr}^b\), a quasi-isomorphism as \(F^{b+1}L'\) is acyclic, and \(0\to\operatorname{gr}^pL'\) for \(p\notin[a,b]\), a quasi-isomorphism by hypothesis. \(\square\)

Proposition 3.5.

  1. For intervals \([a,b]\subset[c,d]\) of \(\mathbf Z\), let \(DF^{[a,b]}(\mathcal A)\) be the Verdier quotient of the triangulated subcategory \(KF^{[a,b]}(\mathcal A)\subset KF(\mathcal A)\) by its filtered acyclic complexes. The functor \(DF^{[a,b]}(\mathcal A)\to DF^{[c,d]}(\mathcal A)\) is fully faithful, with essential image the objects \(X\) with \(\operatorname{gr}^pX\cong0\) for \(p\notin[a,b]\). The same holds for \(DF^{[a,b]}(\mathcal A)\to DF(\mathcal A)\), and \(DF(\mathcal A)\) is the union of these full subcategories.
  2. \(DF^{[0,0]}(\mathcal A)=D(\mathcal A)\). The functor \(\iota\colon D(\mathcal A)\to DF(\mathcal A)\), \(F^0K=K\), \(F^1K=0\), is fully faithful and exact, and the underlying complex of \(\iota K\) is \(K\) and \(\operatorname{gr}\iota K\) is \(K\) placed in degree \(0\) of the grading.
  3. Let \(DF'(\mathcal A)\) be the filtered derived category of [Tag 05RX], built from complexes whose terms have finite filtrations. The inclusion induces a fully faithful exact functor \(DF(\mathcal A)\to DF'(\mathcal A)\) whose essential image consists of the \(X\) with \(\operatorname{gr}^pX\cong0\) for all but finitely many \(p\).

Proof. (1) Let \(L\in CF^{[a,b]}(\mathcal A)\) and \(s\colon L'\to L\) a filtered quasi-isomorphism with \(L'\in CF^{[c,d]}(\mathcal A)\). By Lemma 3.4, with \(L''\) as there, \(s\circ(L''\to L')\) is a filtered quasi-isomorphism in \(KF^{[a,b]}(\mathcal A)\). So in (2.1) the filtered quasi-isomorphisms with source in \(KF^{[a,b]}(\mathcal A)\) are cofinal, and the colimits computing \(\operatorname{Hom}_{DF^{[a,b]}}(L,M)\) and \(\operatorname{Hom}_{DF^{[c,d]}}(L,M)\) agree. Lemma 3.4 also gives the essential image. The same argument applies to \(DF(\mathcal A)\), whose objects all lie in some \(CF^{[a,b]}(\mathcal A)\). (2) A complex with filtration in \([0,0]\) is just a complex, and filtered morphisms, homotopies and quasi-isomorphisms are the ordinary ones. (3) The same argument, Lemma 3.4 being stated for terms with finite filtrations. \(\square\)

By (3), statements proved in the Stacks project for \(DF'(\mathcal A)\), such as the description of \(DF'^+(\mathcal A)\) by complexes of filtered injective objects [Tag 015Q], apply to objects of \(DF(\mathcal A)\). For the intersection of a cohomological bound and a filtration interval we write \(D^{[a,b]}F^{[c,d]}(\mathcal A)=\) \(D^{[a,b]}F(\mathcal A)\cap DF^{[c,d]}(\mathcal A)\), and similarly with \(+\), \(-\), \(b\).

Theorem 3.6. Regard \(\operatorname{Fil}_f(\mathcal A)\) as the full subcategory of \(CF(\mathcal A)\) of complexes concentrated in degree \(0\).

  1. The functor \(\operatorname{Fil}_f(\mathcal A)\to DF(\mathcal A)\) is fully faithful, with essential image \(D^{[0,0]}F(\mathcal A)\). For \(L,M\in\operatorname{Fil}_f(\mathcal A)\) and \(n<0\), \(\operatorname{Ext}^n_{\mathrm{fil}}(L,M)=0\).
  2. Let \(n\ge1\). An \(n\)-extension of \(L\) by \(M\) is a sequence \(E\colon\) \(0\to\) \(M\to\) \(Z_{n-1}\to\) \(\cdots\to\) \(Z_0\to\) \(L\to\) \(0\) of \(\operatorname{Fil}_f(\mathcal A)\) whose associated graded sequence is exact; a morphism of \(n\)-extensions is a commutative diagram which is the identity on \(M\) and \(L\); two \(n\)-extensions are equivalent if they are related by a chain of morphisms. Let \(X_E\) be the filtered complex \(M\to Z_{n-1}\to\cdots\to Z_0\) with \(M\) in degree \(-n\) and \(Z_j\) in degree \(-j\), \(s_E\colon X_E\to L\) the map induced by \(Z_0\to L\), and \(f_E\colon X_E\to M[n]\) the identity of \(M\) in degree \(-n\). Then \(s_E\) is a filtered quasi-isomorphism, and \(E\mapsto\delta(E)=f_E\circ s_E^{-1}\) is a bijection from equivalence classes of \(n\)-extensions onto \(\operatorname{Ext}^n_{\mathrm{fil}}(L,M)\).

Proof. (1) Apply Proposition 3.3 with \(a=b=0\): \(C^{[0,0]}F(\mathcal A)=\operatorname{Fil}_f(\mathcal A)\), and a filtered quasi-isomorphism \(f\) between filtered objects induces isomorphisms \(F^pf\) for all \(p\) (Lemma 2.1), so it is an isomorphism of filtered objects; the localization is \(\operatorname{Fil}_f(\mathcal A)\) itself. For \(n<0\), a morphism \(L\to M[n]\) of \(DF(\mathcal A)\) is \(f s^{-1}\) with \(s\colon L'\to L\) a filtered quasi-isomorphism; replacing \(L'\) by \(\tau_{\leq0}L'\) (Lemma 3.1), \(L'\) vanishes in positive degrees while \(M[n]\) is concentrated in degree \(-n>0\), so \(f=0\).

(2) The cone of \(s_E\) is the sequence \(E\) placed in degrees \(-n-1,\dots,0\); its graded pieces are exact, so \(s_E\) is a filtered quasi-isomorphism (Lemma 2.1). A morphism of \(n\)-extensions \(E\to E'\) gives \(\varphi\colon X_E\to X_{E'}\) with \(s_{E'}\varphi=s_E\) and \(f_{E'}\varphi=f_E\), so \(\delta(E)=\delta(E')\).

Surjectivity. Let \(\xi=fs^{-1}\) with \(s\colon L'\to L\) a filtered quasi-isomorphism and \(f\colon L'\to M[n]\). By Lemma 3.1 we may assume \(L'\) vanishes in positive degrees. Since \(s\) is a filtered quasi-isomorphism, the sequence \(\cdots\to L'^{-1}\to L'^0\to L\to0\) has exact graded pieces, so its maps are strict and \(C=\operatorname{Coker}(L'^{-n-1}\to L'^{-n})\to L'^{-n+1}\) is a strict monomorphism [Tag 05QH]. The component \(f^{-n}\colon L'^{-n}\to M\) kills the image of \(L'^{-n-1}\) and factors through \(C\). Let \(Z_{n-1}=L'^{-n+1}\amalg_CM\) be the pushout in \(\operatorname{Fil}(\mathcal A)\), so \(M\to Z_{n-1}\) is a strict monomorphism with cokernel \(L'^{-n+1}/C\) [Tag 05SM], and \(Z_j=L'^{-j}\) for \(0\le j\le n-2\) (for \(n=1\) only \(Z_0\) occurs). The resulting \(E\colon\) \(0\to\) \(M\to\) \(Z_{n-1}\to\) \(\cdots\to\) \(Z_0\to\) \(L\to\) \(0\) has exact graded pieces: it is spliced from the strict short exact sequence \(0\to M\to Z_{n-1}\to L'^{-n+1}/C\to0\) and the strict exact sequence \(0\to\) \(L'^{-n+1}/C\to\) \(L'^{-n+2}\to\) \(\cdots\to\) \(L'^0\to\) \(L\to\) \(0\) (for \(n=1\), from \(0\to M\to Z_0\to L'^0/C\to0\) and \(L'^0/C\cong L\)). The map \(\psi\colon L'\to X_E\) equal to zero in degrees \(<-n\), to \(f^{-n}\) in degree \(-n\), to \(L'^{-n+1}\to Z_{n-1}\) in degree \(-n+1\) and to the identity in degrees \(>-n+1\) is a morphism of complexes, because \((dx,0)\) and \((0,f^{-n}x)\) have the same image in \(Z_{n-1}\), with \(s_E\psi=s\) and \(f_E\psi=f\); hence \(\delta(E)=f_E s_E^{-1}=fs^{-1}=\xi\).

Injectivity. Suppose \(\delta(E)=\delta(E')\). By the description of morphisms in a localization there are \(L_3\) and filtered quasi-isomorphisms \(t\colon L_3\to X_E\), \(t'\colon L_3\to X_{E'}\) with \(s_Et=s_{E'}t'\) and \(f_Et=f_{E'}t'\) in \(KF(\mathcal A)\); replacing \(L_3\) by \(\tau_{\leq0}L_3\) we may assume \(L_3^q=0\) for \(q>0\), and we choose representatives \(t,t'\). A homotopy between two morphisms \(L_3\to L[0]\) has a single component \(L_3^1\to L\), which is zero; so \(s_Et=s_{E'}t'=:\sigma\) on the nose. A homotopy \(h\) between \(f_Et\) and \(f_{E'}t'\) has a single component \(h\colon L_3^{-n+1}\to M\), and \(t^{-n}-t'^{-n}=h\circ d\) on \(L_3^{-n}\) (the components \(t^{-n},t'^{-n}\) land in \(X_E^{-n}=M=X_{E'}^{-n}\)). Let \(E_3\) be the \(n\)-extension constructed from \((L_3,\sigma,\varphi=f_Et)\) as in the surjectivity part: \(Z^3_{n-1}=(L_3^{-n+1}\oplus M)/\{(dx,-t^{-n}x)\}\). A morphism \(E_3\to E\) is given by \(t^{-j}\) on \(Z^3_j=L_3^{-j}\), \(j\le n-2\), and by \((x,m)\mapsto t^{-n+1}(x)+m\) on \(Z^3_{n-1}\): it is well defined since \((dx,-t^{-n}x)\mapsto\) \(t^{-n+1}(dx)-t^{-n}x=\) \(d(t^{-n}x)-t^{-n}x=\) \(0\), the differential \(X_E^{-n}=M\to Z_{n-1}\) being the inclusion. A morphism \(E_3\to E'\) is given by \(t'^{-j}\) and by \((x,m)\mapsto t'^{-n+1}(x)+m+h(x)\): it is well defined since \(t'(dx)-t^{-n}x+h(dx)=\) \(t'^{-n}x-t^{-n}x+h(dx)=\) \(0\), it commutes with \(Z^3_{n-1}\to Z^3_{n-2}\) because \(M\) maps to zero there, and for \(n=1\) it commutes with the maps to \(L\) because \(s_Et=s_{E'}t'\). All these maps are filtered (\(h\) is a filtered homotopy) and are the identity on \(M\) and \(L\). So \(E\) and \(E'\) are equivalent. \(\square\)

4. Filtered Hom complexes and the filtered derived Hom

Definition 4.1. For \(L,M\in CF(\mathcal A)\), the Hom complex \(\operatorname{Hom}^\bullet(L,M)\) of the underlying complexes is filtered by \[ \begin{aligned}&F^n\operatorname{Hom}^m(L,M)\\&\quad=\{(f^k)_k:\\&\qquad\ f^k(F^pL^k)\subset F^{p+n}M^{k+m}\\&\qquad\ \text{for all }p,k\}\\&\quad=F^0\operatorname{Hom}^m(L,M(n)).\end{aligned} \] The differential preserves this filtration; if \(L\) has filtration in \([a,b]\) and \(M\) in \([c,d]\), then \(F^{c-b}\operatorname{Hom}=\operatorname{Hom}\) and \(F^{d-a+1}\operatorname{Hom}=0\). This defines a functor \(\operatorname{Hom}^\bullet\colon CF(\mathcal A)^{\mathrm{o}}\times CF(\mathcal A)\to CF(\mathrm{Ab})\), and \[ \begin{gathered}\operatorname{Hom}_{CF}(L,M)\\ =Z^0F^0\operatorname{Hom}^\bullet(L,M),\\ \operatorname{Hom}_{KF}(L,M[n])\\ =H^nF^0\operatorname{Hom}^\bullet(L,M),\end{gathered} \tag{4.1} \] since filtered homotopies are the elements of \(F^0\operatorname{Hom}^{-1}\). For \(E,F\in CG(\mathcal A)\), \(\operatorname{Hom}^\bullet(E,F)=\prod_{i,j}\operatorname{Hom}^\bullet(k^iE,k^jF)\) is graded by \(k^n\operatorname{Hom}^\bullet(E,F)=\) \(\prod_{j-i=n}\operatorname{Hom}^\bullet(k^iE,k^jF)\) (finite products). For \(L,M\in CF(\mathcal A)\) the map \[ \begin{aligned}&\gamma\colon\operatorname{gr}\operatorname{Hom}^\bullet(L,M)\\&\quad\longrightarrow\operatorname{Hom}^\bullet(\operatorname{gr}L,\operatorname{gr}M),\\&f\in F^n\mapsto(\operatorname{gr}^pf\colon\\&\qquad\operatorname{gr}^pL\to\operatorname{gr}^{p+n}M)_p,\end{aligned} \tag{4.2} \] is a natural morphism of graded complexes. \(\operatorname{Hom}^\bullet\) is compatible with filtered (resp. graded) homotopies in each variable and defines functors \(KF(\mathcal A)^{\mathrm o}\times KF(\mathcal A)\to KF(\mathrm{Ab})\) and \(KG(\mathcal A)^{\mathrm o}\times KG(\mathcal A)\to KG(\mathrm{Ab})\), exact in each variable: in each variable it carries sequences of \(CF(\mathcal A)\) that are split in each degree (as sequences of filtered objects) to sequences of the same kind, and distinguished triangles are isomorphic to the triangles of such sequences [Tag 014M].

Definition 4.2. \(M\in CF(\mathcal A)\) is of injective type if every \(\operatorname{gr}^pM^k\) is injective; equivalently every \(F^pM^k\) is injective (extensions of injective objects are injective, and a direct summand of an injective object is injective). Then each \(F^{p+1}M^k\subset F^pM^k\) is split, and \(M^k\cong\bigoplus_q M^k_q\) with \(F^pM^k=\bigoplus_{q\ge p}M^k_q\) [Tag 05TP]. \(E\in CG(\mathcal A)\) is of injective type if every \(k^pE^k\) is injective.

Lemma 4.3. If \(M\) is of injective type, \(\gamma\) in (4.2) is an isomorphism.

Proof. Both sides are products over \(k\) of the corresponding constructions for the filtered objects \(L^k,M^{k+m}\) (graded pieces commute with products of abelian groups), so let \(L=X\) and \(M=Y\) be filtered objects, \(Y\) with injective graded pieces. Both sides are additive in \(Y\), so by the splitting of Definition 4.2 we may assume that \(Y\) has a single graded piece, in degree \(q\). Then \(f(F^iX)\subset F^{i+n}Y\) means \(f(F^{q-n+1}X)=0\), so \(F^n\operatorname{Hom}(X,Y)=\operatorname{Hom}(X/F^{q-n+1}X,Y)\). Since \(\operatorname{Hom}(-,Y)\) is exact, the sequence \(0\to\) \(\operatorname{gr}^{q-n}X\to\) \(X/F^{q-n+1}X\to\) \(X/F^{q-n}X\to\) \(0\) gives \(\operatorname{gr}^n\operatorname{Hom}(X,Y)\cong\operatorname{Hom}(\operatorname{gr}^{q-n}X,Y)\), by restriction; this is \(\gamma\), because only \(i=q-n\) contributes to \(\prod_i\operatorname{Hom}(\operatorname{gr}^iX,\operatorname{gr}^{i+n}Y)\). \(\square\)

Lemma 4.4. Let \(L\in CF(\mathcal A)\) and \(M\in CF(\mathcal A)\) of injective type with \(M^k=0\) for \(k\ll0\). If \(L\) or \(M\) is filtered acyclic, \(\operatorname{Hom}^\bullet(L,M)\) is filtered acyclic. The same holds for graded complexes.

Proof. Let \(X\) be a complex and \(I\) a bounded below complex of injectives. If \(X\) is acyclic, every morphism \(X\to I[n]\) is homotopic to zero [Tag 013R], so \(\operatorname{Hom}^\bullet(X,I)\) is acyclic; if \(I\) is acyclic, \(\mathrm{id}_I\) is homotopic to zero by the same lemma, and \(\operatorname{Hom}^\bullet(X,I)\) is contractible. This is the graded case, componentwise. In the filtered case, \(\operatorname{gr}\operatorname{Hom}^\bullet(L,M)\cong\operatorname{Hom}^\bullet(\operatorname{gr}L,\operatorname{gr}M)\) by Lemma 4.3, and \(\operatorname{gr}M\) is a bounded below graded complex of injective type. \(\square\)

Lemma 4.5. Assume \(\mathcal A\) has enough injectives. Let \(M\in CF(\mathcal A)\) have filtration in \([a,b]\) and \(M^k=0\) for \(k<m\). There is a filtered quasi-isomorphism \(f\colon M\to I\) with \(I\) of injective type, filtration in \([a,b]\), \(I^k=0\) for \(k<m\), and every \(\operatorname{gr}^pf\) a monomorphism in each degree.

Proof. We construct monomorphisms of complexes \(f_p\colon F^pM\to I_p\) that are quasi-isomorphisms, with \(I_p\) complexes of injectives vanishing in degrees \(<m\) and \(I_{p+1}\subset I_p\), by descending induction on \(p\). For \(p=b\) use [Tag 013K] (3). Given \(f_p\), let \(J=I_p\amalg_{F^pM}F^{p-1}M\) (pushout of complexes, degreewise). Then \(F^{p-1}M\to J\) is a monomorphism with cokernel \(I_p/F^pM\), which is acyclic, so it is a quasi-isomorphism; and \(I_p\to J\) is a monomorphism with cokernel \(\operatorname{gr}^{p-1}M\). Choose a monomorphism and quasi-isomorphism \(J\to I_{p-1}\) into a complex of injectives vanishing in degrees \(<m\) [Tag 013K], and let \(f_{p-1}\) be the composite \(F^{p-1}M\to J\to I_{p-1}\); the composite \(I_p\to J\to I_{p-1}\) is a monomorphism, through which we regard \(I_p\) as a subcomplex of \(I_{p-1}\); then \(f_{p-1}\) restricts to \(f_p\) on \(F^pM\). Put \(I=I_a\), \(F^pI=I_p\) for \(a\le p\le b\) (and \(F^pI=I\), \(0\) outside), \(f=f_a\). Each \(I_{p+1}^k\subset I_p^k\) is split, so \(\operatorname{gr}^pI^k\) is injective. The morphism of short exact sequences \((F^{p+1}M\to F^pM\to\operatorname{gr}^pM)\to\) \((I_{p+1}\to I_p\to\operatorname{gr}^pI)\) has first two vertical maps quasi-isomorphisms, so \(\operatorname{gr}^pf\) is one; by the snake lemma \(\operatorname{Ker}\operatorname{gr}^pf\) injects into \(\operatorname{Coker}f_{p+1}=I_{p+1}/F^{p+1}M\), which maps isomorphically onto \(J/F^pM\subset I_p/F^pM\) (with \(J\) the pushout of the step from \(p+1\) to \(p\)); so the map \(\operatorname{Coker}f_{p+1}\to\operatorname{Coker}f_p\) is injective and \(\operatorname{gr}^pf\) is a monomorphism. \(\square\)

Proposition 4.6. Assume \(\mathcal A\) has enough injectives.

  1. For \(I\in CF(\mathcal A)\) of injective type and bounded below, and every \(L\in KF(\mathcal A)\), \(\operatorname{Hom}_{KF}(L,I)\to\operatorname{Hom}_{DF}(L,I)\) is bijective.
  2. Let \(K^+FI(\mathcal A)\subset KF(\mathcal A)\) be the full subcategory of bounded below complexes of injective type. The functor \(K^+FI(\mathcal A)\to D^+F(\mathcal A)\) is an exact equivalence.
  3. For \(L\in KF(\mathcal A)\) and \(M\in K^+F(\mathcal A)\), the right derived functor of \(\operatorname{Hom}^\bullet(L,-)\colon K^+F(\mathcal A)\to DF(\mathrm{Ab})\) is defined at \(M\) [Tag 05S9], with value \(\operatorname{Hom}^\bullet(L,I)\) for every filtered quasi-isomorphism \(M\to I\) into a bounded below complex of injective type. This defines a functor \[ \begin{aligned}&R\operatorname{Hom} \colon DF(\mathcal A)^{\mathrm o}\times D^+F(\mathcal A)\\&\quad \longrightarrow DF(\mathrm{Ab}),\end{aligned} \] exact in each variable, with a natural map \(\operatorname{Hom}^\bullet(L,M)\to R\operatorname{Hom}(L,M)\).
  4. For \(L\in DF(\mathcal A)\), \(M\in D^+F(\mathcal A)\) and \(n\in\mathbf Z\), \(\operatorname{Ext}^n_{\mathrm{fil}}(L,M)=\) \(H^nF^0R\operatorname{Hom}(L,M)\).

Proof. (1) If \(C\) is filtered acyclic, \(\operatorname{Hom}^\bullet(C,I)\) is filtered acyclic (Lemma 4.4); its \(F^0\) has a finite filtration with acyclic graded pieces, hence is acyclic, and \(\operatorname{Hom}_{KF}(C,I)=H^0F^0\operatorname{Hom}^\bullet(C,I)=0\) by (4.1). For a filtered quasi-isomorphism \(s\colon L'\to L\), the long exact sequence of \(\operatorname{Hom}_{KF}(-,I)\) [Tag 0149] on the triangle of \(s\), whose third vertex is filtered acyclic, shows that \(\operatorname{Hom}_{KF}(L,I)\to\operatorname{Hom}_{KF}(L',I)\) is bijective. A morphism \(L\to I\) of \(DF(\mathcal A)\) is \(gs^{-1}\) with \(g\colon L'\to I\); writing \(g=g's\) shows surjectivity, and if \(g'\in\operatorname{Hom}_{KF}(L,I)\) maps to zero then \(g's=0\) for some \(s\), so \(g'=0\). (2) \(K^+FI(\mathcal A)\) is stable under shifts and cones, so it is a triangulated subcategory and the functor is exact; it is fully faithful by (1) and essentially surjective by Proposition 3.3 (with \(b=+\infty\)) and Lemma 4.5. (3) Let \(u\colon M\to I\) be a filtered quasi-isomorphism into a bounded below complex of injective type (Lemma 4.5). For every filtered quasi-isomorphism \(t\colon M\to M'\), (1) gives \(g_t\colon M'\to I\) with \(g_tt=u\) in \(KF(\mathcal A)\), unique there; \(g_t\) is a filtered quasi-isomorphism. For a morphism \(v\) from \(t\) to \(t'\) (that is, \(vt=t'\)), \(g_{t'}v\) and \(g_t\) agree after composition with \(t\), hence are equal. So the maps \(\operatorname{Hom}^\bullet(L,g_t)\) form a cocone with vertex \(\operatorname{Hom}^\bullet(L,I)\) over the ind-object \((t\colon M\to M')\mapsto\operatorname{Hom}^\bullet(L,M')\), and the object \(i=(u\colon M\to I)\), with \(g_u=\mathrm{id}\), satisfies the two conditions of [Tag 05PU]: for an object \(j=(t\colon M\to M')\) take \(k=i\) and the morphism \(g_t\colon j\to i\). Hence the derived functor is defined at \(M\) with value \(\operatorname{Hom}^\bullet(L,I)\), as in the proof of [Tag 05TH]. Choosing a quasi-inverse \(j\) of (2), \(R\operatorname{Hom}(L,M)=\operatorname{Hom}^\bullet(L,jM)\) is exact in \(M\); in \(L\) it carries filtered quasi-isomorphisms to filtered quasi-isomorphisms (Lemma 4.4 on cones), hence factors through \(DF(\mathcal A)^{\mathrm o}\), exactly. (4) By (1) and (4.1), \(\operatorname{Ext}^n_{\mathrm{fil}}(L,M)=\) \(\operatorname{Hom}_{KF}(L,I[n])=\) \(H^nF^0\operatorname{Hom}^\bullet(L,I)\). \(\square\)

Proposition 4.7. Assume \(\mathcal A\) has enough injectives, and let \(L\in DF(\mathcal A)\), \(M\in D^+F(\mathcal A)\).

  1. In the graded case the same constructions give \(R\operatorname{Hom}\colon\) \(DG(\mathcal A)^{\mathrm o}\times D^+G(\mathcal A)\to\) \(DG(\mathrm{Ab})\) with \(k^nR\operatorname{Hom}(E,F)=\) \(\prod_iR\operatorname{Hom}(k^iE,k^{i+n}F)\), computed in \(D(\mathcal A)\), and \(\operatorname{Ext}^n_{\mathrm{grad}}(E,F)=\) \(H^nk^0R\operatorname{Hom}(E,F)\).
  2. \(\gamma\) induces a natural isomorphism \(\operatorname{gr}R\operatorname{Hom}(L,M)\to\) \(R\operatorname{Hom}(\operatorname{gr}L,\operatorname{gr}M)\) of \(DG(\mathrm{Ab})\).
  3. The underlying object of \(R\operatorname{Hom}(L,M)\) is \(R\operatorname{Hom}_{\mathcal A}(L,M)\) of the underlying objects, and \(H^nR\operatorname{Hom}(L,M)=\operatorname{Ext}^n_{\mathcal A}(L,M)\).

Proof. (1) Graded complexes of injective type are families of complexes of injectives, and the statements are componentwise. (2) If \(M\to I\) is a filtered quasi-isomorphism into a bounded below complex of injective type, \(\operatorname{gr}M\to\operatorname{gr}I\) is a graded one into a bounded below graded complex of injective type, and \(\gamma\) is an isomorphism of complexes by Lemma 4.3. (3) \(F^aI=I\) is a bounded below complex of injectives, the underlying complex of \(\operatorname{Hom}^\bullet(L,I)\) is the Hom complex of the underlying complexes, and \(H^n\operatorname{Hom}^\bullet(L,I)=\) \(\operatorname{Hom}_{K(\mathcal A)}(L,I[n])=\) \(\operatorname{Hom}_{D(\mathcal A)}(L,M[n])\) [Tag 05TG]. \(\square\)

5. The spectral sequence of a filtered derived Hom

Definition 5.1. For a filtered complex \(X\) and \(p\in\mathbf Z\), the short exact sequence \(0\to\) \(\operatorname{gr}^{p+1}X\to\) \(F^pX/F^{p+2}X\to\) \(\operatorname{gr}^pX\to\) \(0\) defines, by [Tag 0152], a morphism \[ \partial^p_X\colon\operatorname{gr}^pX\longrightarrow\operatorname{gr}^{p+1}X[1] \] of \(D(\mathcal A)\), natural in \(X\in DF(\mathcal A)\). It is the class of the extension of the two graded pieces.

For a short exact sequence \(0\to A\xrightarrow{i}B\xrightarrow{\pi}C\to0\) of complexes with degreewise splittings \(\sigma\colon C\to B\), \(\rho\colon B\to A\) (\(\rho i=\mathrm{id}\), \(\pi\sigma=\mathrm{id}\), \(i\rho+\sigma\pi=\mathrm{id}\)), the morphism of [Tag 0152] is the morphism of complexes \(\rho\circ d_B\circ\sigma\colon C\to A[1]\): indeed \(c\mapsto(\sigma c,-\rho d_B\sigma c)\) is a morphism of complexes \(C\to C(i)\) right inverse to \(q\), so \(-pq^{-1}\) is represented by \(\rho d_B\sigma\).

Theorem 5.2. Assume \(\mathcal A\) has enough injectives. Let \(L\in DF(\mathcal A)\) and \(M\in D^+F(\mathcal A)\). The filtration of \(R\operatorname{Hom}(L,M)\) defines a bounded spectral sequence converging to the Ext groups of the underlying objects, \[ \begin{aligned}&E_1^{p,q} =H^{p+q}(\operatorname{gr}^pR\operatorname{Hom}(L,M))\\&\quad =\prod_i\operatorname{Ext}^{p+q}(\operatorname{gr}^iL,\operatorname{gr}^{i+p}M)\\&\quad \Longrightarrow\operatorname{Ext}^{p+q}(L,M),\end{aligned} \] the abutment carrying a finite filtration, and for \(x=(x_i)_i\in E_1^{p,q}\), with \(n=p+q\), \[ \begin{aligned}&(d_1x)_i=\partial^{i+p}_M\circ x_i\\&\quad+(-1)^{n+1}\,x_{i+1}\circ\partial^i_L,\end{aligned} \tag{5.1} \] an element of \(\operatorname{Ext}^{n+1}(\operatorname{gr}^iL,\operatorname{gr}^{i+p+1}M)\).

Proof. Represent \(L\) by a filtered complex and let \(M\to I\) be as in Lemma 4.5, so \(R\operatorname{Hom}(L,M)=K:=\operatorname{Hom}^\bullet(L,I)\), a complex with a finite filtration. The spectral sequence of the filtered complex \(K\) [Tag 012M] is bounded and converges to \(H^*(K)\) with finite filtrations [Tag 012W]; \(H^n(K)=\operatorname{Ext}^n(L,M)\) and \(E_1^{p,q}=H^{p+q}(\operatorname{gr}^pK)\) is the stated product by Proposition 4.7. The differential \(d_1\) is the connecting map of \(0\to\) \(\operatorname{gr}^{p+1}K\to\) \(F^pK/F^{p+2}K\to\) \(\operatorname{gr}^pK\to\) \(0\) [Tag 012N].

Choose splittings \(I^k=\bigoplus_jI^k_j\), \(F^pI^k=\bigoplus_{j\ge p}I^k_j\) (Definition 4.2), and write \(d_I=\sum_{r\ge0}d_{(r)}\) with \(d_{(r)}(I_j)\subset I_{j+r}\). Then \((I_j,d_{(0)})=\operatorname{gr}^jI\), \(d_{(0)}d_{(1)}+d_{(1)}d_{(0)}=0\), and \(d_{(1)}\colon\operatorname{gr}^jI\to\operatorname{gr}^{j+1}I[1]\) is a morphism of complexes. By the description of the morphism of [Tag 0152] for degreewise split sequences, \(d_{(1)}\) represents the morphism of the sequence \(0\to\operatorname{gr}^{j+1}I\to F^jI/F^{j+2}I\to\operatorname{gr}^jI\to0\), which is \(\partial^j_M\) by naturality, \(M\to I\) inducing quasi-isomorphisms on the three terms. Elements of \(F^pK^n\) are families \(f=(f_j)_j\), \(f_j\colon L\to I_j\) of degree \(n\), with \(f_j(F^{j-p+1}L)=0\); under Lemma 4.3, the class of \(f\) in \(\operatorname{gr}^pK\) has components \(z_i=f_{i+p}|_{F^iL}\), seen on \(\operatorname{gr}^iL\).

Let \(z=(z_i)\) be a cycle of \(\operatorname{gr}^pK^n\), \(z_i\colon\operatorname{gr}^iL\to\operatorname{gr}^{i+p}I[n]\) representing \(x_i\). Since \(I_j\) is injective, extend \(z_{j-p}\) (on \(F^{j-p}L/F^{j-p+1}L\)) to \(f_j\colon L/F^{j-p+1}L\to I_j\); this gives a lift \(f\in F^pK^n\). The class \((d_1x)_i\) is represented by the restriction of \((\mathrm df)_{i+p+1}=\) \(\sum_r d_{(r)}f_{i+p+1-r}-(-1)^nf_{i+p+1}d_L\) to \(F^iL\). There \(f_{i+p+1-r}\) vanishes for \(r\ge2\), \(f_{i+p}\) restricts to \(z_i\), and \(f_{i+p+1}\) restricts to a map \(g_0\colon X\to I_{i+p+1}\) on \(X=F^iL/F^{i+2}L\) extending \(z_{i+1}\) on \(\operatorname{gr}^{i+1}L\). So the representative is \[ d_{(1)}z_i+\bigl(d_{(0)}g_0-(-1)^ng_0d_X\bigr). \] The first term represents \(\partial^{i+p}_M[n]\circ x_i=\partial^{i+p}_M\circ x_i\). The second vanishes on \(\operatorname{gr}^{i+1}L\) and is \(g'\circ\beta\) for a cycle \(g'\colon\operatorname{gr}^iL\to\operatorname{gr}^{i+p+1}I[n+1]\), \(\beta\colon X\to\operatorname{gr}^iL\) the projection. Let \(\alpha\colon\operatorname{gr}^{i+1}L\to X\) be the inclusion and \(q\colon C(\alpha)\to\operatorname{gr}^iL\), \(p\colon C(\alpha)\to\operatorname{gr}^{i+1}L[1]\) the maps of [Tag 0152]. On \(C(\alpha)^k=X^k\oplus(\operatorname{gr}^{i+1}L)^{k+1}\), the map \(H(x,a)=g_0(x)\) of degree \(-1\) into \(\operatorname{gr}^{i+p+1}I[n+1]\) satisfies \[ \begin{aligned}&d_{I[n+1]}H+H d_{C(\alpha)}\\&\quad =(-1)^{n+1}g'\circ q+z_{i+1}[1]\circ p,\end{aligned} \] as one sees on \((x,a)\) using \(g_0\alpha=z_{i+1}\). Hence \(g'q=(-1)^nz_{i+1}[1]p\) in \(K(\mathcal A)\), and \(g'=\) \((-1)^n z_{i+1}[1]\,pq^{-1}=\) \((-1)^{n+1}z_{i+1}[1]\,\partial^i_L\). So the second term represents \((-1)^{n+1}x_{i+1}\circ\partial^i_L\). \(\square\)

Remark 5.3. Applying (5.1) twice, the mixed terms cancel: \((d_1d_1x)_i=\) \(\partial^{i+p+1}_M\circ\partial^{i+p}_M\circ x_i-x_{i+2}\circ\partial^{i+1}_L\circ\partial^i_L\). Composites of consecutive extension classes vanish, \(\partial^{p+1}_X[1]\circ\partial^p_X=0\) (Exercise 6.5), in accordance with \(d_1^2=0\). For \(d_1^2=0\) the sign of the second term of (5.1) has to change with \(n\); its value is the one computed in the proof.

6. Exercises

Exercise 6.1. Let \(X\ne0\) be an object of \(\mathcal A\) and \(L\) the complex \(X\xrightarrow{\mathrm{id}}X\) in degrees \(-1,0\), with \(F^0L=L\), \(F^1L=(0\to X)\), \(F^2L=0\). Show that \(L\) is acyclic but not zero in \(DF(\mathcal A)\), that \(H^q(L)=0\) for all \(q\) while \(L\to\tau_{\geq0}L\) is not a filtered quasi-isomorphism, and that \(\partial^0_L\colon X[1]\to X[1]\) is the identity.

Exercise 6.2. Let \(X\ne0\) and let \(L\) be \(X\xrightarrow{\mathrm{id}}X\) in degrees \(0,1\), with \(F^0L=L\), \(F^1L=(0\to X)\), \(F^2L=0\). Compute \(\tau_{\leq0}L\) and \(\tau_{\leq0}\operatorname{gr}^0L\), and conclude that \(\operatorname{gr}\) does not commute with \(\tau_{\leq0}\). Show that \(H^0(F_1\tau_{\leq0}L)\to H^0(F_1L)\) is not an isomorphism, so that Lemma 1.2 (2) is optimal.

Exercise 6.3. Assume \(\mathcal A\) has enough injectives. Let \(L,M\) be objects of \(\mathcal A\), \(L\) with filtration concentrated in degree \(p\) (\(F^pL=L\), \(F^{p+1}L=0\)) and \(M\) concentrated in degree \(q\). Show that \(\operatorname{Ext}^n_{\mathrm{fil}}(L,M)=\operatorname{Ext}^n_{\mathcal A}(L,M)\) if \(p\le q\) and \(=0\) if \(p>q\).

Exercise 6.4. Assume \(\mathcal A\) has enough injectives. Let \(L\in D(\mathcal A)\), viewed in \(DF(\mathcal A)\) with filtration in \([0,0]\), and \(M\in D^+F(\mathcal A)\) with filtration in \([0,1]\). Show that the spectral sequence of Theorem 5.2 has two nonzero columns, that \(d_1\colon\operatorname{Ext}^q(L,\operatorname{gr}^0M)\to\operatorname{Ext}^{q+1}(L,\operatorname{gr}^1M)\) is composition with \(\partial^0_M\), and that the resulting short exact sequences are the pieces of the long exact sequence of \(\operatorname{Hom}(L,-)\) on the triangle \(\operatorname{gr}^1M\to M\to\operatorname{gr}^0M\xrightarrow{\partial^0_M}\).

Exercise 6.5. Let \(X\) be a filtered complex. Show that \(\partial^{p+1}_X[1]\circ\partial^p_X=0\).

7. Complete solutions

Solution 6.1. \(L\) is the cone of \(\mathrm{id}_X\) shifted, hence acyclic. Its graded pieces are \(\operatorname{gr}^0L=X[1]\) (\(X\) in degree \(-1\)) and \(\operatorname{gr}^1L=X\) in degree \(0\), not acyclic, so \(L\) is not filtered acyclic and \(L\ncong0\) in \(DF(\mathcal A)\) (Proposition 2.2 (4)). \(\tau_{\geq0}L\) is \(\operatorname{Coker}(\mathrm{id})=0\) in degree \(0\), so \(L\to\tau_{\geq0}L=0\) is not a filtered quasi-isomorphism, in accordance with Lemma 3.1, since \(H^{-1}(\operatorname{gr}^0L)\ne0\). In particular \(H^0(F^1L)=X\) (as \(F^1L=(0\to X)\)) while \(H^0(F^1\tau_{\geq0}L)=0\): in degree \(q=n=0\) the map of Lemma 1.2 (1) for \(F^p\) is not an isomorphism. For \(\partial^0_L\), the sequence \(0\to\operatorname{gr}^1L\to L\to\operatorname{gr}^0L\to0\) is degreewise split with \(\sigma\) the identity of \(X\) in degree \(-1\) and \(\rho\) the identity of \(X\) in degree \(0\); by Definition 5.1, \(\partial^0_L\) is represented by \(\rho d\sigma\), the identity of \(X\) in degree \(-1\), that is, the identity of \(X[1]\).

Solution 6.2. \(\tau_{\leq0}L\) is \(\operatorname{Ker}(\mathrm{id})=0\) in degree \(0\), so \(\tau_{\leq0}L=0\) and \(\operatorname{gr}^0\tau_{\leq0}L=0\). But \(\operatorname{gr}^0L=(X\to0)\), with \(\operatorname{gr}^0L^1=F^0L^1/F^1L^1=0\), so \(\tau_{\leq0}\operatorname{gr}^0L=X\ne0\). Next, \(F_1L=L/F^1L=(X\to0)\) has \(H^0=X\), while \(F_1\tau_{\leq0}L=0\); the map \(0\to X\) is not an isomorphism, and here \(q=0=n\) is outside the range \(q\le n-1\) of Lemma 1.2 (2).

Solution 6.3. Let \(M\to I\) be an injective resolution in \(\mathcal A\), with filtration concentrated in degree \(q\); it is a filtered injective resolution of \(M\). A map \(f\colon L^k\to I^{k+n}\) lies in \(F^m\operatorname{Hom}\) when \(f(F^pL)=f(L)\subset F^{p+m}I\), which is \(I\) if \(p+m\le q\) and \(0\) otherwise. So \(F^0\operatorname{Hom}^\bullet(L,I)=\operatorname{Hom}^\bullet(L,I)\) if \(p\le q\) and \(=0\) if \(p>q\), and Proposition 4.6 (4) gives the claim.

Solution 6.4. \(\operatorname{gr}L=L\) in degree \(0\), so \(E_1^{p,q}=\operatorname{Ext}^{p+q}(L,\operatorname{gr}^pM)\), nonzero only for \(p\in\{0,1\}\). In (5.1) only \(i=0\) occurs and \(x_1=0\), so \(d_1x=\partial^0_M\circ x\). With two columns the spectral sequence degenerates at \(E_2\) and gives exact sequences \(0\to E_2^{1,n-1}\to\operatorname{Ext}^n(L,M)\to E_2^{0,n}\to0\), with \[ \begin{aligned}E_2^{1,n-1}&=\operatorname{Coker}(\partial^0_M\circ-\colon\\&\qquad\operatorname{Ext}^{n-1}(L,\operatorname{gr}^0M)\\&\qquad\to\operatorname{Ext}^n(L,\operatorname{gr}^1M))\end{aligned} \] and \[ \begin{aligned}E_2^{0,n}&=\operatorname{Ker}(\partial^0_M\circ-\colon\\&\qquad\operatorname{Ext}^n(L,\operatorname{gr}^0M)\\&\qquad\to\operatorname{Ext}^{n+1}(L,\operatorname{gr}^1M)).\end{aligned} \] The long exact sequence of \(\operatorname{Hom}(L,-[n])\) on the triangle \(\operatorname{gr}^1M\to M\to\operatorname{gr}^0M\xrightarrow{\partial^0_M}\operatorname{gr}^1M[1]\) [Tag 0149] cuts into exactly these short exact sequences, its connecting map being composition with \(\partial^0_M\). The edge maps agree with the maps of that sequence: here \(F^1M=\operatorname{gr}^1M\), \(F^1K=\operatorname{Hom}^\bullet(L,F^1I)\), and the filtration step \(F^1\operatorname{Ext}^n(L,M)\) is the image of \(\operatorname{Ext}^n(L,\operatorname{gr}^1M)\).

Solution 6.5. Let \(Y=F^pX/F^{p+3}X\), with \(Y_1=F^{p+1}X/F^{p+3}X\subset Y\) and \(Y_2=\operatorname{gr}^{p+2}X\subset Y_1\). The morphism of short exact sequences \[ \begin{gathered}(0\to Y_1\to Y\to\operatorname{gr}^pX\to0)\\\longrightarrow(0\to\operatorname{gr}^{p+1}X\to F^pX/F^{p+2}X\\\to\operatorname{gr}^pX\to0),\end{gathered} \] given by \(Y_1\to Y_1/Y_2=\operatorname{gr}^{p+1}X\), \(Y\to Y/Y_2=F^pX/F^{p+2}X\) and the identity of \(\operatorname{gr}^pX\), and the naturality of the morphisms of [Tag 0152] show that \(\partial^p_X\) is the composite of the morphism \(\delta\colon\operatorname{gr}^pX\to Y_1[1]\) of the second sequence with \(Y_1[1]\to\operatorname{gr}^{p+1}X[1]\). In the distinguished triangle \(Y_2\to Y_1\to\operatorname{gr}^{p+1}X\xrightarrow{\partial^{p+1}_X}Y_2[1]\), two consecutive morphisms compose to zero, so \(\partial^{p+1}_X[1]\circ(Y_1[1]\to\operatorname{gr}^{p+1}X[1])=0\) and hence \(\partial^{p+1}_X[1]\circ\partial^p_X=0\).

Sources

The filtered derived category, the filtered derived Hom functor and its spectral sequence are due to Illusie, in his work on the cotangent complex, where they serve to make derived constructions on flags of complexes and to prove the additivity of traces and the Whitney formula for Chern classes of perfect complexes; the reduction of statements on filtered objects to their graded pieces follows his treatment. The AI-integrated Stacks project, Chapter Derived Categories, treats the filtered derived category of an abelian category built from complexes whose terms have finite filtrations, and resolutions by filtered injective objects. Derived and triangulated categories are due to Verdier, and the description of Ext groups by extensions to Yoneda.