Filtered tensor products on a ringed topos

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

Let \(A\) be a commutative ring of a topos \(T\). The tensor product of two filtered complexes of \(A\)-modules is filtered by the sums of the images of the products of filtration steps. This lesson derives that tensor product on the filtered derived category \(DF(A)\) of Filtered derived categories, for complexes without any boundedness condition, and proves:

Prerequisites. Filtered derived categories (Lemma 2.1, Propositions 2.2, 2.3, 3.3, 3.5, Definition 4.2 and Lemma 4.3). From the AI-integrated Stacks project: modules on a site form a Grothendieck abelian category [Stacks, Tag 07A5]; flat modules and short exact sequences [Tag 03EX], [Tag 03EY], [Tag 05VD]; K-flat complexes [Tag 06YN], [Tag 06YP], [Tag 06YQ], [Tag 06YS], [Tag 077J], [Tag 07A2], [Tag 0G7B], [Tag 0G7E]; derived pullback [Tag 06YV], [Tag 07A4]; tor amplitude [Tag 08FZ], [Tag 08G1]; K-injective complexes [Tag 070I], [Tag 090X] and their filtered version [Tag 0BKI]; localization of a triangulated category through a subcategory [Tag 0GSL]; morphisms in a localization [Tag 05Q4]; symmetric monoidal categories and their functors [Tag 0FFK], [Tag 0FFW], [Tag 0FFY].

Conventions

\(T\) is a topos, identified with the category of sheaves on a site, and \(A\) a commutative ring of \(T\); \(\operatorname{Mod}(A)\) is a Grothendieck abelian category [Tag 07A5]. We write \(CF(A)\), \(KF(A)\), \(DF(A)\), \(CG(A)\), \(DG(A)\), \(D(A)\) for the categories of the lesson on filtered derived categories with \(\mathcal A=\operatorname{Mod}(A)\), and \(\otimes\) for \(\otimes_A\). The tensor product of complexes is the total complex \((X\otimes Y)^n=\bigoplus_{i+j=n}X^i\otimes Y^j\) with \(d(x\otimes y)=dx\otimes y+(-1)^{|x|}x\otimes dy\) on homogeneous local sections. Statements about local sections are statements about sheaves: two morphisms out of a tensor product agree when they agree on local sections of the form \(x\otimes y\).

1. Tensor products of filtered complexes

Definition 1.1. For \(L,M\in CF(A)\), the complex \(L\otimes M\) is filtered by \[ \begin{aligned}&F^n(L\otimes M)\\&\quad =\sum_{p+q=n}\operatorname{Im}(F^pL\otimes F^qM\to L\otimes M).\end{aligned} \] If \(L\) has filtration in \([a,b]\) and \(M\) in \([c,d]\), \(L\otimes M\) has filtration in \([a+c,b+d]\). For graded complexes, \(k^n(E\otimes F)=\bigoplus_{p+q=n}k^pE\otimes k^qF\). The composite \(F^pL\otimes F^qM\to\) \(F^{p+q}(L\otimes M)\to\) \(\operatorname{gr}^{p+q}(L\otimes M)\) vanishes on the images of \(F^{p+1}L\otimes F^qM\) and \(F^pL\otimes F^{q+1}M\) and defines a natural morphism of graded complexes \[ \begin{aligned}&\mu \colon\operatorname{gr}L\otimes\operatorname{gr}M\\&\quad \longrightarrow\operatorname{gr}(L\otimes M).\end{aligned} \tag{1.1} \]

Lemma 1.2. On complexes of \(A\)-modules, the isomorphisms \(\alpha\colon(X\otimes Y)\otimes Z\to X\otimes(Y\otimes Z)\), \((x\otimes y)\otimes z\mapsto x\otimes(y\otimes z)\), \(\sigma\colon X\otimes Y\to Y\otimes X\), \(x\otimes y\mapsto(-1)^{|x||y|}y\otimes x\), and \(\lambda\colon A\otimes X\to X\), \(a\otimes x\mapsto ax\), are isomorphisms of complexes, and of filtered (resp. graded) complexes when \(X,Y,Z\) are filtered (resp. graded). Let \(X_1,\dots,X_m\) be complexes and let \(\phi\) be any composite of these isomorphisms, their inverses, and their tensor products with identities, from one iterated tensor product of \(X_1,\dots,X_m\) (with factors \(A\) inserted anywhere) to another. If \(\phi\) moves the factors according to the permutation \(\pi\), then \[ \begin{gathered}\phi(x_1\otimes\cdots\otimes x_m)\\ =\varepsilon\,x_{\pi(1)}\otimes\cdots\otimes x_{\pi(m)},\\ \varepsilon\\ =\prod_{i<j,\ \pi^{-1}(i)>\pi^{-1}(j)}(-1)^{|x_i||x_j|},\end{gathered} \] with brackets and unit factors suppressed. In particular two such composites with the same source, target and permutation are equal; this is the coherence of the symmetric monoidal category of complexes, of filtered complexes and of graded complexes, and \(\mu\) of (1.1) is compatible with \(\alpha\), \(\sigma\) and \(\lambda\).

Proof. The three maps are compatible with the differentials (for \(\sigma\): \(d\sigma(x\otimes y)=\) \((-1)^{|x||y|}(dy\otimes x+(-1)^{|y|}y\otimes dx)\) and \(\sigma d(x\otimes y)=\) \((-1)^{(|x|+1)|y|}y\otimes\) \(dx+\) \((-1)^{|x|}(-1)^{|x|(|y|+1)}dy\otimes\) \(x\), which agree), and they carry \(F^n\) onto \(F^n\) and \(k^n\) onto \(k^n\). Each of them, tensored with identities, acts on a product of homogeneous local sections by a permutation of the factors and the sign \(\prod(-1)^{|x_i||x_j|}\) over the pairs whose order it reverses. For a composite, a pair reversed twice contributes \((-1)^{2|x_i||x_j|}=1\), so the composite acts by the permutation \(\pi\) and the product over the pairs reversed by \(\pi\). For \(\mu\), both \(\mu\circ(\text{structure map})\) and \((\text{structure map})\circ\mu\) act on \(\bar x\otimes\bar y\) by the same formula. \(\square\)

Definition 1.3. \(L\in CF(A)\) has flat graded pieces if every \(\operatorname{gr}^pL^k\) is a flat \(A\)-module. Then every \(F^pL^k\), \(L^k/F^pL^k\) and \(L^k\) is flat [Tag 03EY].

Lemma 1.4. Let \(L,M\in CF(A)\), and assume \(L\) has flat graded pieces. Then (1.1) is an isomorphism, and every map \(F^pL\otimes F^qM\to L\otimes M\) is injective.

Proof. The second assertion: \(F^pL\otimes F^qM\to F^pL\otimes M\) is injective because the terms of \(F^pL\) are flat, and \(F^pL\otimes M\to L\otimes M\) because the terms of \(L/F^pL\) are flat [Tag 03EX]. For the first, both sides are, in degree \(n\), direct sums over \(i+j=n\) of the corresponding constructions for the filtered modules \(L^i,M^j\), since \(F^r\) of a direct sum of filtered modules is the direct sum. So let \(X\) be a filtered module with flat graded pieces and filtration in \([a,b]\), and \(Y\) a filtered module; we argue by induction on \(b-a\).

If \(a=b\), then \(F^r(X\otimes Y)=X\otimes F^{r-a}Y\), and since \(X\) is flat, \(\operatorname{gr}^r(X\otimes Y)=\) \(X\otimes\operatorname{gr}^{r-a}Y=\) \((\operatorname{gr}X\otimes\operatorname{gr}Y)^r\).

If \(a<b\), let \(X'=F^{a+1}X\), with \(F^pX'=F^{\max(p,a+1)}X\), and \(X''=X/X'=\operatorname{gr}^aX\), with filtration concentrated in degree \(a\). Since \(X''\) is flat, \(0\to X'\otimes Y\to X\otimes Y\to X''\otimes Y\to0\) is exact. We show that it is exact on each \(F^n\). Surjectivity: \(F^n(X''\otimes Y)=X''\otimes F^{n-a}Y\) is the image of \(X\otimes F^{n-a}Y=\) \(\operatorname{Im}(F^aX\otimes F^{n-a}Y)\subset\) \(F^n(X\otimes Y)\). Next, \[ \begin{gathered}F^n(X\otimes Y)\\ =\operatorname{Im}(X\otimes F^{n-a}Y)+S,\\ S\\ =\sum_{p\ge a+1}\operatorname{Im}(F^pX\otimes F^{n-p}Y)\\ =F^n(X'\otimes Y),\end{gathered} \] because for \(p\le a+1\) the term \(\operatorname{Im}(F^pX'\otimes F^{n-p}Y)\) lies in \(\operatorname{Im}(F^{a+1}X\otimes F^{n-a-1}Y)\). As \(S\subset X'\otimes Y\), \(F^n(X\otimes Y)\cap(X'\otimes Y)=\) \((\operatorname{Im}(X\otimes F^{n-a}Y)\cap(X'\otimes Y))+S\). A local section of \(X\otimes F^{n-a}Y\) whose image lies in \(X'\otimes Y\) maps to zero in \(X''\otimes Y\), hence to zero in \(X''\otimes F^{n-a}Y\) (which injects into \(X''\otimes Y\), \(X''\) being flat), hence comes from \(X'\otimes F^{n-a}Y\), whose image lies in \(\operatorname{Im}(F^{a+1}X\otimes F^{n-a-1}Y)\subset S\). So \(F^n(X\otimes Y)\cap(X'\otimes Y)=F^n(X'\otimes Y)\), and \[ \begin{aligned}&0 \to F^n(X'\otimes Y) \to F^n(X\otimes Y)\\&\quad \to F^n(X''\otimes Y) \to0\end{aligned} \] is exact for every \(n\); by the snake lemma so is \(0\to\) \(\operatorname{gr}^n(X'\otimes Y)\to\) \(\operatorname{gr}^n(X\otimes Y)\to\) \(\operatorname{gr}^n(X''\otimes Y)\to\) \(0\). The maps \(\mu\) form a morphism to it from the split exact sequence \(0\to\) \((\operatorname{gr}X'\otimes\operatorname{gr}Y)^n\to\) \((\operatorname{gr}X\otimes\operatorname{gr}Y)^n\to\) \((\operatorname{gr}X''\otimes\operatorname{gr}Y)^n\to\) \(0\) (\(\operatorname{gr}X=\operatorname{gr}X'\oplus\operatorname{gr}X''\)); the outer maps are isomorphisms by induction and by the case \(a=b\), so the middle one is. \(\square\)

Without flatness the lemma fails (Exercise 5.1).

2. Filtered K-flat and K-injective resolutions

Definition 2.1. \(P\in CF(A)\) is filtered K-flat if every \(\operatorname{gr}^pP\) is a K-flat complex [Tag 06YN] whose terms are flat. \(I\in CF(A)\) is filtered K-injective if every \(\operatorname{gr}^pI\) is a K-injective complex whose terms are injective. Graded complexes are graded K-flat or graded K-injective when their components are K-flat with flat terms, or K-injective with injective terms.

Lemma 2.2. Let \(P,Q\) be filtered K-flat and \(M\in CF(A)\).

  1. \(P\), \(F^pP\) and \(P/F^pP\) are K-flat with flat terms. Filtered K-flat complexes are stable under shifts and cones.
  2. \(P\otimes Q\) is filtered K-flat, and \(\operatorname{gr}(P\otimes Q)\cong\operatorname{gr}P\otimes\operatorname{gr}Q\).
  3. If \(M\) or \(P\) is filtered acyclic, \(P\otimes M\) is filtered acyclic. Consequently \(P\otimes u\) is a filtered quasi-isomorphism for every filtered quasi-isomorphism \(u\colon M\to M'\), and \(v\otimes M\) is one for every filtered quasi-isomorphism \(v\colon P\to P'\) between filtered K-flat complexes.
  4. If \(I\) is filtered K-injective, then \(I\), \(F^pI\) and \(I/F^pI\) are K-injective with injective terms, the filtration of each \(I^k\) splits, and filtered K-injective complexes are stable under shifts and cones.

Proof. (1) By induction on the length of the filtration, from the sequences \(0\to F^{p+1}P\to F^pP\to\operatorname{gr}^pP\to0\) and \(0\to\operatorname{gr}^pP\to P/F^{p+1}P\to P/F^pP\to0\), whose third terms have flat terms [Tag 0G7B], [Tag 03EY]. Cones: \(\operatorname{gr}C(f)=C(\operatorname{gr}f)\), and cones of K-flat complexes with flat terms are K-flat [Tag 0G7B]. (2) Lemma 1.4 and [Tag 07A2]. (3) By Lemma 1.4, \(\operatorname{gr}^n(P\otimes M)\cong\bigoplus_{p+q=n}\operatorname{gr}^pP\otimes\operatorname{gr}^qM\), a finite sum. If \(\operatorname{gr}^qM\) is acyclic, \(\operatorname{gr}^pP\otimes\operatorname{gr}^qM\) is acyclic as \(\operatorname{gr}^pP\) is K-flat; if \(\operatorname{gr}^pP\) is acyclic, \(0\to\operatorname{gr}^pP\) is a quasi-isomorphism of K-flat complexes and \(\operatorname{gr}^pP\otimes\operatorname{gr}^qM\) is acyclic [Tag 06YP]. The consequences follow by applying this to cones. (4) Extensions of injective modules split, so \(F^{p+1}I^k\to F^pI^k\) splits and all these terms are injective; \(F^pI\) and \(I/F^pI\) are iterated extensions of the \(\operatorname{gr}^qI\) split in each degree, hence K-injective [Tag 090X]; the same argument applies to cones. \(\square\)

Proposition 2.3. Let \(M\in CF(A)\) have filtration in \([a,b]\).

  1. There is a filtered quasi-isomorphism \(P\to M\), surjective in each degree, with \(P\) filtered K-flat with filtration in \([a,b]\).
  2. There is a filtered quasi-isomorphism \(M\to I\) with \(I\) filtered K-injective with filtration in \([a,b]\).

Proof. (1) Write \(M_p=M/F^pM\), so \(M_a=0\), \(M_{b+1}=M\), and \(0\to\operatorname{gr}^pM\to M_{p+1}\to M_p\to0\). We construct complexes \(P_p\), \(a\le p\le b+1\), K-flat with flat terms, quasi-isomorphisms \(\varepsilon_p\colon P_p\to M_p\) surjective in each degree, and surjections \(\pi_p\colon P_{p+1}\to P_p\) with \(\varepsilon_p\pi_p=(M_{p+1}\to M_p)\varepsilon_{p+1}\). Put \(P_a=0\). Given \(P_p\), let \(X=P_p\times_{M_p}M_{p+1}\) (fibre product of complexes, degreewise). The projection \(X\to P_p\) is surjective with kernel \(\operatorname{gr}^pM\), and \(X\to M_{p+1}\) is surjective, because \(\varepsilon_p\) is; comparing \(0\to\operatorname{gr}^pM\to X\to P_p\to0\) with \(0\to\operatorname{gr}^pM\to M_{p+1}\to M_p\to0\), the five lemma shows that \(X\to M_{p+1}\) is a quasi-isomorphism. Choose a K-flat complex \(P_{p+1}\) with flat terms and a quasi-isomorphism \(P_{p+1}\to X\) surjective in each degree [Tag 06YS], and let \(\varepsilon_{p+1}\), \(\pi_p\) be its composites with the two projections. The kernel \(G_p\) of \(\pi_p\) has flat terms [Tag 03EY] and is K-flat [Tag 0G7B]; comparing \(0\to G_p\to P_{p+1}\to P_p\to0\) with \(0\to\operatorname{gr}^pM\to X\to P_p\to0\) shows that the induced map \(G_p\to\operatorname{gr}^pM\) is a quasi-isomorphism. Put \(P=P_{b+1}\) and \(F^pP=\operatorname{Ker}(P\to P_p)\). Then \(F^aP=P\), \(F^{b+1}P=0\), \(\operatorname{gr}^pP\cong G_p\), \(\varepsilon_{b+1}\colon P\to M\) maps \(F^pP\) into \(F^pM\), and \(\operatorname{gr}^p\varepsilon_{b+1}\) is the quasi-isomorphism \(G_p\to\operatorname{gr}^pM\).

(2) By [Tag 0BKI] there is a morphism of filtered complexes \(j\colon M\to J\) such that all \(F^pJ^n/F^{p'}J^n\) are injective, all \(F^pJ/F^{p'}J\) are K-injective, and \(j\) induces quasi-isomorphisms \(F^pM/F^{p'}M\to F^pJ/F^{p'}J\) for \(p\le p'\) (including \(p'=+\infty\)). Let \(I\) be the complex \(F^aJ\) with \(F^pI=F^aJ\) for \(p\le a\), \(F^pI=F^pJ\) for \(a\le p\le b\) and \(F^pI=0\) for \(p>b\). Then \(\operatorname{gr}^pI=F^pJ/F^{p+1}J\) for \(a\le p<b\) and \(\operatorname{gr}^bI=F^bJ\) are K-injective with injective terms, and \(j\) maps \(M=F^aM\) into \(F^aJ\), filtered, with \(\operatorname{gr}^pM\to\operatorname{gr}^pI\) a quasi-isomorphism for all \(p\) (for \(p=b\), \(\operatorname{gr}^bM=F^bM\to F^bJ\)). \(\square\)

Corollary 2.4. Let \(KF_{\mathrm{fl}}(A)\) and \(KF_{\mathrm{inj}}(A)\) be the full subcategories of \(KF(A)\) of filtered K-flat and of filtered K-injective complexes.

  1. They are triangulated subcategories, and \(KF_{\mathrm{fl}}(A)[FQis^{-1}]\to DF(A)\) is an exact equivalence.
  2. For \(I\) filtered K-injective and every \(L\in KF(A)\), \(\operatorname{Hom}_{KF}(L,I)=\operatorname{Hom}_{DF}(L,I)\); hence \(KF_{\mathrm{inj}}(A)\to DF(A)\) is an exact equivalence.

The same holds for graded complexes.

Proof. (1) Lemma 2.2 and Proposition 2.3 (1) with [Tag 0GSL]. (2) For a filtered acyclic \(C\), \(\operatorname{gr}\operatorname{Hom}^\bullet(C,I)\cong\operatorname{Hom}^\bullet(\operatorname{gr}C,\operatorname{gr}I)\) (Lemma 4.3 of the lesson on filtered derived categories, \(I\) being of injective type), which is acyclic since each \(\operatorname{gr}^qI\) is K-injective [Tag 070I]; so \(F^0\operatorname{Hom}^\bullet(C,I)\), having a finite filtration with acyclic graded pieces, is acyclic, and \(\operatorname{Hom}_{KF}(C,I)=H^0F^0\operatorname{Hom}^\bullet(C,I)=0\). The argument of Proposition 4.6 (1) of that lesson then gives the bijection, and Proposition 2.3 (2) gives essential surjectivity. \(\square\)

3. The derived tensor product and the symmetric monoidal structure

Lemma 3.1. Let \(S\) be a multiplicative system in a category \(\mathcal C\), and \(Q\colon\mathcal C\to S^{-1}\mathcal C\) the localization. Then \(Q\times Q\colon\mathcal C\times\mathcal C\to S^{-1}\mathcal C\times S^{-1}\mathcal C\) is a localization at \(S\times S\). For every category \(\mathcal D\), composition with \(Q\) (resp. \(Q\times Q\)) is fully faithful on functor categories, with image the functors inverting \(S\) (resp. \(S\times S\)).

Proof. \(S\times S\) is a multiplicative system in \(\mathcal C\times\mathcal C\), and by [Tag 05Q4] the morphisms in \((S\times S)^{-1}(\mathcal C\times\mathcal C)\) are colimits over the product of the two filtered categories of fractions of the products of the morphism sets, which are the products of the colimits. A natural transformation \(G\circ Q\to G'\circ Q\) is natural with respect to every \(Q(f)\) and every \(Q(s)^{-1}\), hence with respect to all morphisms \(Q(s)^{-1}Q(f)\) of \(S^{-1}\mathcal C\). \(\square\)

Proposition 3.2. Tensor product of complexes induces an exact functor in each variable \[ \begin{gathered}\otimes^{\mathbf L} \colon DF(A)\times DF(A)\\ \longrightarrow DF(A),\\ L\otimes^{\mathbf L}M\cong P\otimes M\cong P\otimes Q,\end{gathered} \] for filtered K-flat resolutions \(P\to L\), \(Q\to M\). The underlying complex of \(L\otimes^{\mathbf L}M\) is the derived tensor product of the underlying complexes, and \(\mu\) of (1.1) gives a natural isomorphism \[ \begin{gathered}\operatorname{gr}(L\otimes^{\mathbf L}M)\\ \cong\operatorname{gr}L\otimes^{\mathbf L}\operatorname{gr}M,\\ \operatorname{gr}^n(L\otimes^{\mathbf L}M)\\ \cong\bigoplus_{p+q=n}\operatorname{gr}^pL\otimes^{\mathbf L}\operatorname{gr}^qM.\end{gathered} \tag{3.1} \] For \(L,M\in KF(A)\), the pro-object \((L'\to L,M'\to M)\mapsto L'\otimes M'\), over pairs of filtered quasi-isomorphisms, is essentially constant with value \(P\otimes Q\); so \(\otimes^{\mathbf L}\) is the left derived functor of \(\otimes\).

Proof. By Lemma 2.2, \(\otimes\colon KF_{\mathrm{fl}}(A)\times KF_{\mathrm{fl}}(A)\to KF_{\mathrm{fl}}(A)\) carries \(FQis\times FQis\) to \(FQis\); with Corollary 2.4 (1) and Lemma 3.1 it induces \(\otimes^{\mathbf L}\). Exactness in each variable holds on \(KF_{\mathrm{fl}}(A)\), where cones are preserved. The maps \(P\otimes Q\to P\otimes M\) are filtered quasi-isomorphisms (Lemma 2.2 (3)). The underlying complex of \(P\) is K-flat with flat terms (Lemma 2.2 (1)), so \(P\otimes M\) also computes the derived tensor product of the underlying complexes; \(\operatorname{gr}P\to\operatorname{gr}L\) is a graded K-flat resolution and \(\mu\colon\operatorname{gr}P\otimes\operatorname{gr}Q\to\operatorname{gr}(P\otimes Q)\) is an isomorphism (Lemma 1.4). For the last assertion, the pairs \((P'\to L,Q'\to M)\) with \(P',Q'\) filtered K-flat are initial among all pairs (Proposition 2.3 (1) applied to \(L'\) and \(M'\)), and all transition maps between their values \(P'\otimes Q'\) are filtered quasi-isomorphisms (Lemma 2.2 (3)); hence the pro-object is essentially constant. \(\square\)

Theorem 3.3. The isomorphisms \(\alpha\), \(\sigma\), \(\lambda\) of Lemma 1.2 on filtered K-flat complexes induce natural isomorphisms \[ \begin{gathered}(L\otimes^{\mathbf L}M)\otimes^{\mathbf L}N\\ \cong L\otimes^{\mathbf L}(M\otimes^{\mathbf L}N),\\ L\otimes^{\mathbf L}M\cong M\otimes^{\mathbf L}L,\\ A\otimes^{\mathbf L}L\cong L\end{gathered} \] making \((DF(A),\otimes^{\mathbf L},A)\) a symmetric monoidal category [Tag 0FFW]; any two composites of these isomorphisms, their inverses and their tensor products with identities, between the same iterated tensor products and inducing the same permutation of the factors, are equal. The same holds for \(D(A)\) (filtration in \([0,0]\)) and for \(DG(A)\). The functor "underlying complex" \(DF(A)\to D(A)\), with the identity comparison maps, and \(\operatorname{gr}\colon DF(A)\to DG(A)\), with the comparison (3.1), are functors of symmetric monoidal categories [Tag 0FFY]. Under the equivalence of \(DG(A)\) with finitely supported families of objects of \(D(A)\), \((E\otimes^{\mathbf L}F)_n=\bigoplus_{p+q=n}E_p\otimes^{\mathbf L}F_q\), and the symmetry of \(DG(A)\) is the sum of the symmetries \(E_p\otimes^{\mathbf L}F_q\cong F_q\otimes^{\mathbf L}E_p\) of \(D(A)\), without any sign depending on \(p,q\).

Proof. On \(KF_{\mathrm{fl}}(A)\), the tensor product with \(\alpha\), \(\sigma\), \(\lambda\) is a symmetric monoidal structure with unit \(A\) (filtered K-flat with filtration in \([0,0]\)), all coherence diagrams commuting by Lemma 1.2. By Lemma 3.1 the functors and natural isomorphisms descend to \(KF_{\mathrm{fl}}(A)[FQis^{-1}]\), equivalent to \(DF(A)\) (Corollary 2.4), and equalities between natural transformations descend as well; this gives the structure and its coherence. For the underlying complex, a filtered K-flat complex is K-flat with flat terms and the structure maps are literally those of complexes. For \(\operatorname{gr}\), the isomorphisms \(\mu\colon\operatorname{gr}P\otimes\operatorname{gr}Q\to\operatorname{gr}(P\otimes Q)\) of Lemma 1.4 are compatible with \(\alpha\), \(\sigma\), \(\lambda\) by Lemma 1.2, and the unit \(A\) has \(\operatorname{gr}A=A\) in degree \(0\). The description of \(DG(A)\) follows from Proposition 2.3 of the lesson on filtered derived categories and from the definition of the tensor product of graded complexes, whose symmetry is the Koszul sign of the cohomological degrees only. \(\square\)

Remark 3.4. Twists are compatible with the product: \(L(m)\otimes^{\mathbf L}M(n)=(L\otimes^{\mathbf L}M)(m+n)\), the filtrations of the two sides being equal on filtered K-flat models; similarly in \(DG(A)\). A morphism \(E\to F(n)\) of \(DG(A)\) has components \(E_p\to F_{p+n}\).

4. Derived pullback and finite tor dimension

Proposition 4.1. Let \(f\colon(T',A')\to(T,A)\) be a morphism of ringed topoi.

  1. For \(P\) filtered K-flat over \(A\), the subcomplexes \(f^*F^pP\subset f^*P\) form a filtration with \(\operatorname{gr}^pf^*P=f^*\operatorname{gr}^pP\), and \(f^*P\) is filtered K-flat over \(A'\).
  2. \(Lf^*L=f^*P\), for \(P\to L\) a filtered K-flat resolution, defines an exact functor \(Lf^*\colon DF(A)\to DF(A')\), compatible with the underlying complexes (with the derived pullback of [Tag 06YV]) and with \(\operatorname{gr}\): \(\operatorname{gr}Lf^*L\cong Lf^*\operatorname{gr}L\), componentwise.
  3. \(Lf^*\) is a functor of symmetric monoidal categories: \(Lf^*(L\otimes^{\mathbf L}M)\cong Lf^*L\otimes^{\mathbf L}Lf^*M\) and \(Lf^*A=A'\), compatibly with \(\alpha\), \(\sigma\), \(\lambda\), and with (3.1) after \(\operatorname{gr}\).

Proof. (1) Since \(\operatorname{gr}^pP\) has flat terms, \(Lf^*\operatorname{gr}^pP^k=f^*\operatorname{gr}^pP^k\) has no cohomology in degree \(-1\); the long exact sequence of \(Lf^*\) on \(0\to F^{p+1}P^k\to F^pP^k\to\operatorname{gr}^pP^k\to0\) shows that \(f^*F^{p+1}P^k\to f^*F^pP^k\) is injective with cokernel \(f^*\operatorname{gr}^pP^k\). These are K-flat with flat terms [Tag 0G7E], [Tag 05VD]. (2) By (1) and Lemma 2.2 (3), \(f^*\) carries filtered quasi-isomorphisms between filtered K-flat complexes to filtered quasi-isomorphisms (it does so on graded pieces, where it computes \(Lf^*\) of [Tag 06YV]); Corollary 2.4 (1) gives the functor, exact as \(f^*\) commutes with cones. (3) On filtered K-flat complexes \(f^*(P\otimes Q)=f^*P\otimes f^*Q\) as filtered complexes, the identification acting on local sections by \(x\otimes y\mapsto x\otimes y\); it commutes with \(\alpha\), \(\sigma\), \(\lambda\) and with \(\mu\) [Tag 07A4]. \(\square\)

Proposition 4.2. Assume every \(A\)-module has tor amplitude in \([-d,0]\) for some \(d\ge0\) [Tag 08FZ].

  1. If \(L\in D^{\geq a}F(A)\) and \(M\in D^{\geq b}F(A)\), then \(L\otimes^{\mathbf L}M\in D^{\geq a+b-d}F(A)\). In particular \(\otimes^{\mathbf L}\) maps \(D^+F\times D^+F\) into \(D^+F\) and \(D^bF\times D^bF\) into \(D^bF\).
  2. For a morphism \(f\colon(T',A')\to(T,A)\), \(Lf^*\) maps \(D^{\geq a}F(A)\) into \(D^{\geq a-d}F(A')\), hence \(D^+F\) into \(D^+F\) and \(D^bF\) into \(D^bF\).

Proof. By (3.1) and Proposition 4.1 (2) it suffices to treat \(D(A)\). (1) Let \(K\in D^{\geq a}(A)\), \(N\in D^{\geq b}(A)\), and \(Q\to N\) a K-flat resolution. Then \(K\otimes Q=\operatorname{colim}_n\tau_{\leq n}K\otimes Q\) and \(H^i\) commutes with filtered colimits, so \(H^i(K\otimes^{\mathbf L}N)=\operatorname{colim}_nH^i(\tau_{\leq n}K\otimes^{\mathbf L}N)\). By the triangles \(\tau_{\leq m-1}K\to\tau_{\leq m}K\to H^m(K)[-m]\), it suffices to see that \(H^m(K)[-m]\otimes^{\mathbf L}N\in D^{\geq m+b-d}\) for \(m\ge a\). Represent \(H^m(K)\) by a complex \(F\) of flat modules in degrees \([-d,0]\) [Tag 08G1] and \(N\) by a complex vanishing in degrees \(<b\) (Proposition 3.3 of the lesson on filtered derived categories); \(F\) is K-flat [Tag 06YQ], so \(F\otimes N\), which vanishes in degrees \(<b-d\), computes \(H^m(K)\otimes^{\mathbf L}N\). (2) Let \(N\in D^{\geq a}(A)\). By [Tag 077J] there are bounded above complexes \(K_n\) of flat modules with quasi-isomorphisms \(K_n\to\tau_{\leq n}N\), forming a direct system with \(\operatorname{colim}K_n\to N\) a K-flat resolution; so \(H^i(Lf^*N)=\) \(\operatorname{colim}_nH^i(f^*K_n)=\) \(\operatorname{colim}_nH^i(Lf^*\tau_{\leq n}N)\). As in (1) it suffices that \(Lf^*G\in D^{\geq-d}\) for a module \(G\), which holds as \(G\) is represented by a complex of flat modules in degrees \([-d,0]\) [Tag 08G1] and \(f^*\) of it computes \(Lf^*G\). \(\square\)

The bound is sharp (Exercise 5.4).

5. Exercises

Exercise 5.1. Let \(T\) be the punctual topos and \(A=\mathbf Z\). Let \(X=\mathbf Z\) with \(F^0X=X\), \(F^1X=2\mathbf Z\), \(F^2X=0\), and \(Y=\mathbf Z/2\) with filtration in \([0,0]\). Show that \(\mu\colon\operatorname{gr}X\otimes\operatorname{gr}Y\to\operatorname{gr}(X\otimes Y)\) is not injective.

Exercise 5.2. Let \(0\to L'\to L\to L''\to0\) be a short exact sequence of complexes of \(A\)-modules, and filter \(L\) by \(F^0L=L\), \(F^1L=L'\), \(F^2L=0\). For \(N\in D(A)\), placed in \(DF(A)\) with filtration in \([0,0]\), show that \(L\otimes^{\mathbf L}N\) has graded pieces \(L''\otimes^{\mathbf L}N\) and \(L'\otimes^{\mathbf L}N\), and that the triangle \(L'\otimes^{\mathbf L}N\to L\otimes^{\mathbf L}N\to L''\otimes^{\mathbf L}N\to\) obtained from the filtration is the tensor product with \(N\) of the triangle of the short exact sequence.

Exercise 5.3. Let \(E\in DG(A)\) be concentrated in weight \(1\), with \(E_1=K\). Show that \(E\otimes^{\mathbf L}E\) is concentrated in weight \(2\) with \((E\otimes^{\mathbf L}E)_2=K\otimes^{\mathbf L}K\), that its symmetry is the symmetry of \(K\otimes^{\mathbf L}K\) in \(D(A)\), and that \(E\cong A(-1)\otimes^{\mathbf L}\iota K\), where \(\iota K\) is \(K\) in weight \(0\).

Exercise 5.4. Let \(T\) be the punctual topos and \(A=\mathbf Z\), so \(d=1\). Show that \(\mathbf Z/2\otimes^{\mathbf L}\mathbf Z/2\) lies in \(D^{\geq-1}\) but not in \(D^{\geq0}\).

6. Complete solutions

Solution 5.1. \(\operatorname{gr}^0X=\mathbf Z/2\) and \(\operatorname{gr}^1X=2\mathbf Z\cong\mathbf Z\), so \(\operatorname{gr}X\otimes\operatorname{gr}Y\) is \(\mathbf Z/2\) in weight \(0\) and \(2\mathbf Z\otimes\mathbf Z/2\cong\mathbf Z/2\) in weight \(1\). On the other side \(X\otimes Y=\mathbf Z/2\) and \(F^1(X\otimes Y)\) is the image of \(2\mathbf Z\otimes\mathbf Z/2\to\mathbf Z\otimes\mathbf Z/2\), \(2\otimes1\mapsto2\otimes1=0\); so \(\operatorname{gr}^1(X\otimes Y)=0\), and \(\mu\) is zero on the nonzero weight-one part. Here \(\operatorname{gr}^0X\) is not flat.

Solution 5.2. Represent \(N\) by a K-flat complex \(Q\) with flat terms, with filtration in \([0,0]\); it is filtered K-flat, so \(L\otimes^{\mathbf L}N\cong L\otimes Q\) (Proposition 3.2, resolving the second factor), filtered by \(F^1=L'\otimes Q\), a subcomplex since \(Q\) has flat terms. Its graded pieces are \(L''\otimes Q\) and \(L'\otimes Q\), which compute \(L''\otimes^{\mathbf L}N\) and \(L'\otimes^{\mathbf L}N\). The triangle of a filtration with two steps is the triangle of the short exact sequence \(0\to F^1\to F^0\to\operatorname{gr}^0\to0\), here \(0\to L'\otimes Q\to L\otimes Q\to L''\otimes Q\to0\), the tensor product with \(Q\) of the given sequence; and the triangle of a short exact sequence of complexes is natural, so tensoring the triangle of \(0\to L'\to L\to L''\to0\) with the K-flat \(Q\) gives the same triangle.

Solution 5.3. By Theorem 3.3, \((E\otimes^{\mathbf L}E)_n=\bigoplus_{p+q=n}E_p\otimes^{\mathbf L}E_q\), which is \(K\otimes^{\mathbf L}K\) for \(n=2\) and \(0\) otherwise, and the symmetry is that of \(D(A)\) on this summand. The graded object \(A(-1)\) has \(A\) in weight \(1\) (\(k^p(A(-1))=k^{p-1}A\)); so \(A(-1)\otimes^{\mathbf L}\iota K\) has \(A\otimes^{\mathbf L}K\cong K\) in weight \(1\).

Solution 5.4. \(\mathbf Z/2\) is represented by \(\mathbf Z\xrightarrow2\mathbf Z\) in degrees \(-1,0\), a complex of flat modules, so \(\mathbf Z/2\otimes^{\mathbf L}\mathbf Z/2\) is \(\mathbf Z/2\xrightarrow0\mathbf Z/2\), with \(H^{-1}=\mathbf Z/2\ne0\) and \(H^0=\mathbf Z/2\). Every abelian group has tor amplitude in \([-1,0]\), and Proposition 4.2 gives \(D^{\geq-1}\).

Sources

The filtered derived tensor product, the computation of its graded pieces and the reduction of filtered statements to graded ones are due to Illusie, in his work on the cotangent complex, where the tensor product is derived on bounded above filtered complexes by flat resolutions. Resolutions of unbounded complexes by K-flat and K-injective complexes are due to Spaltenstein; the AI-integrated Stacks project, Chapters Cohomology on Sites and Injectives, constructs them for modules on a ringed site and for filtered complexes of a Grothendieck abelian category. The sign rule for the symmetry of the tensor product of complexes is Koszul's.