# Filtered and graded powers of complexes

*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 \(T\), and let \(\mathbf T\) be one of the functors \(S\), \(\Lambda\), \(\Gamma\) of symmetric, exterior and divided powers. A filtration of an \(A\)-module \(L\) induces a filtration of every power \(\mathbf T^n(L)\): a product of \(n\) elements lies in filtration degree \(k\) when the degrees of its factors add up to at least \(k\). This lesson studies these filtrations and their derived versions:

- the graded pieces of \(\mathbf T^n(L)\) receive a natural surjection from the powers of the graded module \(\operatorname{gr}L\), and this map is an isomorphism when the graded pieces of \(L\) are flat; for a filtration of length two this is the Koszul filtration of a power of an extension;
- the construction extends to filtered and graded complexes and derives to functors \(L\mathbf T^n\) on the bounded above filtered and graded derived categories, compatible with the underlying complexes and with twists;
- the graded pieces of \(L\mathbf T^n(L)\) are the sums of the derived tensor products \(L\mathbf T^{\nu_1}(\operatorname{gr}^{i_1}L)\otimes\) \(^{\mathbf L}\cdots\otimes\) \(^{\mathbf L}L\mathbf T^{\nu_r}(\operatorname{gr}^{i_r}L)\) with \(\sum\nu_j=n\) and \(\sum\nu_ji_j\) the weight;
- the products \(L\mathbf T^m\otimes^{\mathbf L}L\mathbf T^n\to L\mathbf T^{m+n}\) are filtered, and \(L\mathbf T^n\) multiplies tor amplitude and perfect amplitude intervals containing \(0\) by \(n\).

**Prerequisites.** [*Non-additive functors of complexes in a ringed topos*](non-additive-functors-of-complexes-in-a-ringed-topos.md) (Example 1.3, Section 2.2, Propositions 2.2 and 2.3, Section 3 with Theorem 3.2, Lemmas 4.1 and 4.2, Theorem 4.4, Proposition 4.6, Example 4.7 and Section 5); [*Filtered derived categories*](filtered-derived-categories.md) (Definition 1.1, Lemma 2.1, Lemma 3.1, Propositions 3.3 and 3.5); [*Filtered tensor products on a ringed topos*](filtered-tensor-products-on-a-ringed-topos.md) (Definitions 1.1, 1.3 and 2.1, Lemma 1.4, Propositions 2.3 and 3.2); [*Perfect filtered complexes and duality*](perfect-filtered-complexes-and-duality.md) (Definition 1.1); [*Koszul complexes and the shift formulas*](koszul-complexes-and-the-shift-formulas.md) (Theorem 3.2, in Exercise 3.4 only). From the AI-integrated Stacks project: bounded above resolutions [[Stacks, Tag 05T7](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-subcategory-left-resolution)], every module is a quotient of a flat module [[Tag 03EW](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-modules.html#sites-modules-lemma-module-quotient-flat)], flatness in short exact sequences [[Tag 03EY](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-modules.html#sites-modules-lemma-flat-ses)], bounded above complexes of flat modules are K-flat [[Tag 06YQ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-cohomology.html#sites-cohomology-lemma-bounded-flat-K-flat)], tor amplitude and flat representatives [[Tag 08G1](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-cohomology.html#sites-cohomology-lemma-tor-amplitude)], localization of a triangulated subcategory [[Tag 0GSL](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-localization-subcategory)] and morphisms in a localization as colimits [[Tag 05Q4](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/categories.html#categories-remark-right-localization-morphisms-colimit)].

## Conventions

\(T\) is a topos, identified with the category of sheaves on a site, and \(A\) a commutative ring of \(T\). Modules are \(A\)-modules and \(\otimes=\otimes_A\). 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 \(D^-F(A)\), \(D^{\leq c}F(A)\) for the full subcategories of Definition 3.2 there. As in the lesson on filtered tensor products, statements about local sections are statements about sheaves: the constructions below are made on sections and sheafified, and sheafification is exact and commutes with tensor products, images and quotients.

\(\mathbf T=\bigoplus_{n\geq0}\mathbf T^n\) is one of the functors \(S\), \(\Lambda\), \(\Gamma\) of Example 1.3 of the lesson on non-additive functors, with \(\mathbf T^0(M)=A\) and \(\mathbf T^1(M)=M\); divided powers are presented as in the conventions of the lesson on Koszul complexes. For a map \(f\) of modules, \(\mathbf T(f)\) is a homomorphism of graded algebras. The algebra \(\mathbf T(M)\) is commutative for \(S\) and \(\Gamma\); for \(\Lambda\), \(xy=(-1)^{mn}yx\) for \(x\in\Lambda^m(M)\), \(y\in\Lambda^n(M)\). The letter \(\Gamma\) also denotes the inverse of normalization, as in the mixed objects \(\Gamma\,iX\) of Section 2.2 of the lesson on non-additive functors; the context distinguishes the two.

*Multi-indices.* A multi-index is a family \(\nu=(\nu_i)_{i\in\mathbf Z}\) of nonnegative integers, almost all zero. Put \(|\nu|=\sum_i\nu_i\) and \(w(\nu)=\sum_ii\nu_i\), the *weight*; \(\delta_i\) has \(1\) in place \(i\) and \(0\) elsewhere. For modules \(M_a,\ldots,M_b\) with maps \(M_i\to M\) and \(\nu\) supported in \([a,b]\), let
\[
m_\nu\colon\bigotimes_{i=a}^b\mathbf T^{\nu_i}(M_i)\longrightarrow\mathbf T^{|\nu|}(M)
\]
be the product, in increasing order of \(i\), of the maps \(\mathbf T^{\nu_i}(M_i)\to\mathbf T^{\nu_i}(M)\); factors with \(\nu_i=0\) are \(\mathbf T^0=A\). For \(M=\bigoplus_{i=a}^bM_i\), Example 1.3 of the lesson on non-additive functors gives the *exponential isomorphism*
\[
\begin{aligned}&\sum_{|\nu|=n}m_\nu\\&\quad \colon\bigoplus_{|\nu|=n}\ \bigotimes_{i=a}^b\mathbf T^{\nu_i}(M_i)\\&\quad \xrightarrow{\ \sim\ }\mathbf T^n\Bigl(\bigoplus_{i=a}^bM_i\Bigr),\end{aligned} \tag{0.1}
\]
natural in the modules \(M_i\). For two summands it is the isomorphism of algebras \(\mathbf T(M_1)\otimes\mathbf T(M_2)\cong\mathbf T(M_1\oplus M_2)\), where for \(\Lambda\) the tensor product of algebras carries the sign rule.

*Filtered and graded modules.* Filtered modules have finite filtrations (Definition 1.1 of the lesson on filtered derived categories); the twist \(L(m)\) has \(F^p(L(m))=F^{p+m}L\). A graded module is a module \(P=\bigoplus_ik^iP\) with finitely many nonzero components, and \(k^p(P(m))=k^{p+m}P\). The graded module \(\operatorname{gr}L\) has components \(\operatorname{gr}^iL\), and we say that \(\operatorname{gr}L\) is flat when every \(\operatorname{gr}^iL\) is. A graded module \(P\) defines the filtered module \(P^{\mathrm{fil}}\) with \(F^kP^{\mathrm{fil}}=\bigoplus_{i\geq k}k^iP\), and \(\operatorname{gr}(P^{\mathrm{fil}})=P\). Tensor products of filtered modules carry the filtration of Definition 1.1 of the lesson on filtered tensor products, and \(\mu\colon\operatorname{gr}X\otimes\operatorname{gr}Y\to\operatorname{gr}(X\otimes Y)\) denotes the map (1.1) there.

## 1. Powers of filtered and graded modules

**Definition 1.1.** Let \(L\) be a filtered module with filtration in \([a,b]\), and \(n\geq0\). For \(k\in\mathbf Z\), let \(F^k\mathbf T^n(L)\) be the sum of the images of the maps
\[
m_\nu\colon\bigotimes_{i=a}^b\mathbf T^{\nu_i}(F^iL)\longrightarrow\mathbf T^n(L)
\]
over the multi-indices \(\nu\) supported in \([a,b]\) with \(|\nu|=n\) and \(w(\nu)\geq k\). For a graded module \(P\) with \(k^iP=0\) for \(i\notin[a,b]\), let \(k^p\mathbf T^n(P)\) be the image under (0.1), for \(M_i=k^iP\), of the sum of the summands with \(w(\nu)=p\); thus \(\mathbf T^n(P)=\bigoplus_pk^p\mathbf T^n(P)\).

For \(S\) and \(\Lambda\), \(\mathbf T^{\nu_i}(F^iL)\) is a quotient of \((F^iL)^{\otimes\nu_i}\), so \(F^k\mathbf T^n(L)\) is the image of \(F^k(L^{\otimes n})\) under \(L^{\otimes n}\to\mathbf T^n(L)\): replacing an index \(i<a\) by \(a\) does not change \(F^iL=L\) and raises the weight. For \(\Gamma\), the module \(\Gamma^m(M)\) is generated by the products \(\gamma_{m_1}(x_1)\cdots\gamma_{m_r}(x_r)\) with \(m_1+\cdots+m_r=m\); so \(F^k\Gamma^n(L)\) is generated by the products \(\gamma_{n_1}(x_1)\cdots\gamma_{n_r}(x_r)\) with \(x_j\in F^{i_j}L\), \(\sum_jn_j=n\) and \(\sum_jn_ji_j\geq k\).

**Lemma 1.2.** Let \(L\), \(M\) be filtered modules with filtrations in \([a,b]\), and \(n\geq0\).

1. \(F^k\mathbf T^n(L)\) does not depend on the choice of \([a,b]\). The filtration of \(\mathbf T^n(L)\) lies in \([na,nb]\); \(\mathbf T^0(L)=A\) has filtration in \([0,0]\), and \(\mathbf T^1(L)=L\) carries the filtration of \(L\). If the filtration of \(L\) lies in \([a,a]\), that of \(\mathbf T^n(L)\) lies in \([na,na]\).
2. A filtered map \(f\colon L\to M\) maps \(F^k\mathbf T^n(L)\) into \(F^k\mathbf T^n(M)\).
3. \(F^p\mathbf T^m(L)\cdot F^q\mathbf T^n(L)\subset\) \(F^{p+q}\mathbf T^{m+n}(L)\).
4. \(\mathbf T^n(L(m))=\mathbf T^n(L)(nm)\) as filtered modules.
5. The exponential isomorphism \(\Phi\colon\) \(\bigoplus_{p+q=n}\mathbf T^p(L)\otimes\mathbf T^q(M)\to\) \(\mathbf T^n(L\oplus M)\) is an isomorphism of filtered modules, the summands of the source carrying the tensor filtrations.
6. For graded modules \(P\), \(Q\): \(k^p\mathbf T^m(P)\cdot k^q\mathbf T^n(P)\subset\) \(k^{p+q}\mathbf T^{m+n}(P)\); graded maps induce graded maps; \(\mathbf T^n(P(m))=\mathbf T^n(P)(nm)\); the exponential isomorphism for \(P\oplus Q\) is graded; and \(\mathbf T^n(P^{\mathrm{fil}})=\mathbf T^n(P)^{\mathrm{fil}}\).
7. The modules \(F^k\mathbf T^n(L)\) are compatible with restriction to the objects of \(T\) and commute with local inductive limits of local systems of filtered modules with filtrations in a fixed interval \([a,b]\).

**Proof.** (1) Enlarging \([a,b]\) to \([a',b']\) adds the multi-indices with \(\nu_i>0\) for some \(i\notin[a,b]\). If \(i>b\), then \(F^iL=0\) and \(\mathbf T^{\nu_i}(0)=0\), so \(m_\nu=0\). If \(i<a\), then \(F^iL=L=F^aL\), and the image of \(m_\nu\) lies in that of \(m_{\nu'}\), \(\nu'=\nu+\nu_i(\delta_a-\delta_i)\), because \(\mathbf T^{\nu_a}(F^aL)\cdot\mathbf T^{\nu_i}(F^aL)\) lies in the image of \(\mathbf T^{\nu_a+\nu_i}(F^aL)\); and \(w(\nu')=w(\nu)+\nu_i(a-i)>w(\nu)\). So the two sums agree. For \(k\leq na\) the multi-index \(n\delta_a\) has weight \(na\geq k\) and \(m_{n\delta_a}\) is the identity of \(\mathbf T^n(F^aL)=\mathbf T^n(L)\); for \(k>nb\) no multi-index supported in \([a,b]\) with \(|\nu|=n\) has weight \(\geq k\). For \(n=0\) the only multi-index is \(0\), of weight \(0\); for \(n=1\) they are the \(\delta_i\), and \(F^k\mathbf T^1(L)=\sum_{i\geq k}F^iL=F^kL\). If \(a=b\) the only multi-index is \(n\delta_a\).

(2) \(\mathbf T^{\nu_i}(f)\) maps \(\mathbf T^{\nu_i}(F^iL)\) into \(\mathbf T^{\nu_i}(F^iM)\), compatibly with the maps \(m_\nu\).

(3) For multi-indices \(\nu\), \(\nu'\), the product of the images of \(m_\nu\) and \(m_{\nu'}\) lies in the image of \(m_{\nu+\nu'}\): regroup the factors by their index \(i\) (for \(\Lambda\) the regrouping only changes signs) and use \(\mathbf T^{\nu_i}(F^iL)\cdot\mathbf T^{\nu'_i}(F^iL)\subset\mathbf T^{\nu_i+\nu'_i}(F^iL)\) in \(\mathbf T(F^iL)\). Weights add.

(4) \(F^i(L(m))=F^{i+m}L\). The substitution \(\nu'_j=\nu_{j-m}\) gives \(w(\nu')=w(\nu)+nm\) for \(|\nu|=n\), so \(F^k\mathbf T^n(L(m))=F^{k+nm}\mathbf T^n(L)\).

(5) By (2) for the inclusions of \(L\) and \(M\) and by (3), \(\Phi\) maps the image of \(F^i\mathbf T^p(L)\otimes F^j\mathbf T^q(M)\) into \(F^{i+j}\mathbf T^n(L\oplus M)\). Conversely, let \(|\nu|=n\) and \(w(\nu)\geq k\). Since \(F^i(L\oplus M)=F^iL\oplus F^iM\), the exponential isomorphism for this sum shows that \(\mathbf T^{\nu_i}(F^i(L\oplus M))\) is the sum of the images of \(\mathbf T^{\alpha_i}(F^iL)\otimes\mathbf T^{\beta_i}(F^iM)\) with \(\alpha_i+\beta_i=\nu_i\). Regrouping factors, the image of \(m_\nu\) is the sum, over the decompositions \(\nu=\alpha+\beta\) into multi-indices, of the images under \(\Phi\) of \(\operatorname{Im}(m_\alpha)\otimes\operatorname{Im}(m_\beta)\), the maps \(m_\alpha\) and \(m_\beta\) being those of Definition 1.1 for \(L\) and for \(M\). These lie in \(F^{w(\alpha)}\mathbf T^{|\alpha|}(L)\otimes F^{w(\beta)}\mathbf T^{|\beta|}(M)\) with \(w(\alpha)+w(\beta)=w(\nu)\geq k\). So \(\Phi\) maps \(F^k\) of the source onto \(F^k\mathbf T^n(L\oplus M)\).

(6) The first four statements follow from (0.1) and its naturality: weights add under products and a twist by \(m\) changes the weight of every factor by \(m\). For \(P^{\mathrm{fil}}\), \(F^iP^{\mathrm{fil}}=\bigoplus_{j\geq i}k^jP\), so by (0.1) for this sum \(\mathbf T^{\nu_i}(F^iP^{\mathrm{fil}})\) is the sum of the summands \(\bigotimes_j\mathbf T^{\rho_j}(k^jP)\) with \(|\rho|=\nu_i\) and \(\rho\) supported in \([i,\infty)\). Hence the image of \(m_\nu\) is a sum of summands of (0.1) for \(P\) of weight \(\geq w(\nu)\). Conversely a summand \(\bigotimes_j\mathbf T^{\nu_j}(k^jP)\) of weight \(\geq k\) lies in the image of \(m_\nu\), since \(k^jP\subset F^jP^{\mathrm{fil}}\). So \(F^k\mathbf T^n(P^{\mathrm{fil}})=\bigoplus_{p\geq k}k^p\mathbf T^n(P)\).

(7) \(\mathbf T^n\) commutes with restriction and with local inductive limits (Section 4 of the lesson on non-additive functors), and so do tensor products, finite sums and images (Section 3 there). For a local system \((L_\lambda)\) with local inductive limit \(L\), \(F^iL\) is the local inductive limit of the \(F^iL_\lambda\), by exactness. \(F^k\mathbf T^n(L)\) is a finite sum of images of maps built from these operations. \(\square\)

**Lemma 1.3.** Let \(N\subset M\) be modules and \(n\geq1\). The map \(\mathbf T^n(M)\to\mathbf T^n(M/N)\) is surjective, and its kernel is
\[
\sum_{j=1}^n\operatorname{Im}\bigl(\mathbf T^j(N)\otimes\mathbf T^{n-j}(M)\to\mathbf T^n(M)\bigr),
\]
the map being induced by \(N\subset M\) and the product. For \(S\) and \(\Lambda\) the kernel is the image of \(N\otimes\mathbf T^{n-1}(M)\).

**Proof.** We argue on sections; sheafification is exact. Let \(I\subset\mathbf T(M)\) be the ideal generated by \(N\) for \(S\) and \(\Lambda\), and by the elements \(\gamma_j(y)\), \(y\in N\), \(j\geq1\), for \(\Gamma\). The part of degree \(n\) of \(I\) is the sum in the statement: \(\mathbf T^j(N)\) is spanned by products of elements of \(N\), respectively of divided powers \(\gamma_{j'}(y)\) of elements \(y\in N\) with \(j'\geq1\), so each \(\operatorname{Im}(\mathbf T^j(N)\cdot\mathbf T^{n-j}(M))\) lies in \(I\); conversely \(I\) is spanned by the products \(y\cdot z\), respectively \(\gamma_j(y)\cdot z\), with \(z\) in \(\mathbf T(M)\). For \(S\) and \(\Lambda\), this also shows that \(\operatorname{Im}(N\cdot\mathbf T^{n-1}(M))\) is the whole of \(I\) in degree \(n\).

The map \(\mathbf T(M)\to\mathbf T(M/N)\) is surjective, because \(\mathbf T(M/N)\) is generated as an algebra by the images of elements of \(M\), respectively by the \(\gamma_k(\bar x)\), and it kills \(I\). We construct an inverse of the induced map \(\mathbf T(M)/I\to\mathbf T(M/N)\). For \(S\) and \(\Lambda\), the map \(M\to\mathbf T^1(M)/I^1\) kills \(N\), and the universal property of the symmetric, respectively exterior, algebra extends it to an algebra homomorphism \(\mathbf T(M/N)\to\mathbf T(M)/I\). For \(\Gamma\), send \(\gamma_k(\bar x)\) to the class of \(\gamma_k(x)\) for a lift \(x\in M\) of \(\bar x\). This does not depend on the lift: for \(y\in N\), \(\gamma_k(x+y)=\) \(\gamma_k(x)+\sum_{j\geq1}\gamma_{k-j}(x)\gamma_j(y)\equiv\) \(\gamma_k(x)\bmod I\). The defining relations of \(\Gamma(M/N)\) hold for these classes, since they hold in \(\Gamma(M)\), so this defines an algebra homomorphism \(\Gamma(M/N)\to\Gamma(M)/I\). In each case both composites are algebra homomorphisms that fix generators, so they are identities. \(\square\)

**Construction 1.4.** Let \(L\) have filtration in \([a,b]\), let \(n\geq0\), and let \(\nu\) be a multi-index supported in \([a,b]\) with \(|\nu|=n\). The projections \(F^iL\to\operatorname{gr}^iL\) induce a map
\[
q_\nu\colon\bigotimes_{i=a}^b\mathbf T^{\nu_i}(F^iL)\longrightarrow\bigotimes_{i=a}^b\mathbf T^{\nu_i}(\operatorname{gr}^iL).
\]
By Lemma 1.3 and right exactness of the tensor product, \(q_\nu\) is surjective, and its kernel is the sum of the images of the tensor products in which one factor \(\mathbf T^{\nu_i}(F^iL)\) is replaced by \(\operatorname{Im}(\mathbf T^j(F^{i+1}L)\otimes\mathbf T^{\nu_i-j}(F^iL))\) for some \(j\geq1\). The map \(m_\nu\) sends such a term into the image of \(m_{\nu'}\) with \(\nu'=\nu+j(\delta_{i+1}-\delta_i)\), of weight \(w(\nu)+j\) (the term is zero for \(i=b\)). So the composite of \(m_\nu\) with \(F^{w(\nu)}\mathbf T^n(L)\to\operatorname{gr}^{w(\nu)}\mathbf T^n(L)\) factors uniquely through \(q_\nu\), by a map
\[
c_\nu\colon\bigotimes_{i=a}^b\mathbf T^{\nu_i}(\operatorname{gr}^iL)\longrightarrow\operatorname{gr}^{w(\nu)}\mathbf T^n(L).
\]
Through (0.1) for \(\operatorname{gr}L=\bigoplus_i\operatorname{gr}^iL\), the \(c_\nu\) with \(|\nu|=n\) define a homomorphism of graded modules
\[
\begin{aligned}c_L\colon&\mathbf T^n(\operatorname{gr}L)\\&\longrightarrow\operatorname{gr}\mathbf T^n(L).\end{aligned}
\tag{1.1}
\]

**Proposition 1.5.** Let \(L\), \(M\) be filtered modules.

1. \(c_L\) is surjective, natural in \(L\), compatible with twists, and the identity for \(n\leq1\).
2. \(c_L\) is multiplicative for the products of \(\mathbf T(\operatorname{gr}L)\) and of \(\operatorname{gr}\mathbf T(L)\), the latter induced by Lemma 1.2 (3). For \(S\) and \(\Lambda\), \(c_L\) is the unique homomorphism of bigraded algebras that is the identity on \(\operatorname{gr}L\); for \(\Gamma\), \(c_L(\gamma_k(\bar x))\) is the class of \(\gamma_k(x)\) in \(\operatorname{gr}^{ki}\Gamma^k(L)\) for \(x\in F^iL\) with image \(\bar x\in\operatorname{gr}^iL\).
3. With the exponential isomorphisms \(\Phi\) for \((L,M)\) and \(\Phi_{\operatorname{gr}}\) for \((\operatorname{gr}L,\operatorname{gr}M)\),
\[
\begin{aligned}&c_{L\oplus M}\\&\quad =\operatorname{gr}(\Phi)\circ\mu\circ(c_L\otimes c_M)\circ\Phi_{\operatorname{gr}}^{-1}.\end{aligned}
\]

**Proof.** (1) By Definition 1.1, \(F^k\mathbf T^n(L)\) is the sum of \(F^{k+1}\mathbf T^n(L)\) and of the images of the \(m_\nu\) with \(w(\nu)=k\); so \(\operatorname{gr}^k\mathbf T^n(L)\) is the sum of the images of the \(c_\nu\) with \(w(\nu)=k\). A filtered map commutes with the \(q_\nu\) and \(m_\nu\), which gives naturality; twists follow from Lemma 1.2 (4) and the substitution in its proof. For \(n=0\), \(c_L\) is the identity of \(A\); for \(n=1\), \(c_{\delta_i}\) is the identity of \(\operatorname{gr}^iL\).

(2) For multi-indices \(\nu\), \(\nu'\) and local sections \(\alpha\) of \(\bigotimes_i\mathbf T^{\nu_i}(F^iL)\), \(\beta\) of \(\bigotimes_i\mathbf T^{\nu'_i}(F^iL)\), let \(\alpha\star\beta\) be the local section of \(\bigotimes_i\mathbf T^{\nu_i+\nu'_i}(F^iL)\) obtained by multiplying the factors of equal index, with the sign of the regrouping for \(\Lambda\). Then \(m_\nu(\alpha)\,m_{\nu'}(\beta)=m_{\nu+\nu'}(\alpha\star\beta)\) and \(q_{\nu+\nu'}(\alpha\star\beta)=q_\nu(\alpha)\star q_{\nu'}(\beta)\), and the maps \(\bar m_\nu\) of \(\operatorname{gr}L\) satisfy the same identity. Hence
\[
\begin{aligned}&\quad c_L\bigl(\bar m_\nu(q_\nu\alpha)\,\bar m_{\nu'}(q_{\nu'}\beta)\bigr)\\ &\quad =c_{\nu+\nu'}\bigl(q_\nu\alpha\star q_{\nu'}\beta\bigr)\\ &\quad \quad =\bigl[m_{\nu+\nu'}(\alpha\star\beta)\bigr]\\ &\quad =\bigl[m_\nu(\alpha)\bigr]\,\bigl[m_{\nu'}(\beta)\bigr],\end{aligned}
\]
brackets denoting classes in \(\operatorname{gr}\mathbf T(L)\). The elements \(\bar m_\nu(q_\nu\alpha)\) span \(\mathbf T(\operatorname{gr}L)\) by (0.1) and the surjectivity of \(q_\nu\), so \(c_L\) is multiplicative. \(S(\operatorname{gr}L)\) and \(\Lambda(\operatorname{gr}L)\) are generated by their part of degree \(1\). \(\Gamma(\operatorname{gr}L)\cong\bigotimes_i\Gamma(\operatorname{gr}^iL)\) is generated by the \(\gamma_k(\bar x)\) with \(\bar x\) homogeneous, and \(c_{k\delta_i}(\gamma_k(\bar x))\) is the class of \(m_{k\delta_i}(\gamma_k(x))=\gamma_k(x)\).

(3) Both sides are maps \(\mathbf T(\operatorname{gr}L\oplus\operatorname{gr}M)\to\operatorname{gr}\mathbf T(L\oplus M)\). They are multiplicative: \(c\) by (2); \(\Phi\) and \(\Phi_{\operatorname{gr}}\) are isomorphisms of algebras; and \(\mu(\bar x\otimes\bar y)\,\mu(\bar x'\otimes\bar y')\) is the class of \((x\otimes y)(x'\otimes y')\), with the same sign as in \((\bar x\otimes\bar y)(\bar x'\otimes\bar y')\). They agree on the generators: on \(\operatorname{gr}L\) and \(\operatorname{gr}M\) in degree \(1\) for \(S\) and \(\Lambda\), and on the \(\gamma_k(\bar x)\), \(\bar x\) homogeneous in \(\operatorname{gr}L\) or in \(\operatorname{gr}M\), for \(\Gamma\), both of which go to the class of \(\gamma_k(x)\) in \(\Gamma^k(L\oplus M)\). \(\square\)

**Proposition 1.6.** Let \(L\) be a filtered module such that \(\operatorname{gr}L\) is flat. Then \(c_L\colon\mathbf T^n(\operatorname{gr}L)\to\operatorname{gr}\mathbf T^n(L)\) is an isomorphism for every \(n\), and the modules \(\mathbf T^n(L)\), \(F^k\mathbf T^n(L)\) and \(\operatorname{gr}^k\mathbf T^n(L)\) are flat.

**Proof.** Let the filtration of \(L\) lie in \([a,b]\); we argue by induction on \(b-a\). If \(a=b\), then \(\operatorname{gr}L=\operatorname{gr}^aL=L\) in weight \(a\), the filtration of \(\mathbf T^n(L)\) lies in \([na,na]\) (Lemma 1.2 (1)), and \(c_L=c_{n\delta_a}\) is the identity of \(\mathbf T^n(L)\).

Let \(a<b\). Put \(L'=F^{a+1}L\), with \(F^pL'=F^{\max(p,a+1)}L\), so that \(L'\) has filtration in \([a+1,b]\) and \(\operatorname{gr}L'=\bigoplus_{p>a}\operatorname{gr}^pL\), and put \(E=\operatorname{gr}^aL\). By the theorem of Deligne and Lazard (Theorem 3.2 of the lesson on non-additive functors), \(E\) is the local inductive limit of a local system of finite free modules \(E_\lambda\) over objects \(U_\lambda\), with maps \(E_\lambda\to E|U_\lambda\). Let \(L_\lambda=L|U_\lambda\times_{E|U_\lambda}E_\lambda\), filtered by \(F^pL_\lambda=L_\lambda\) for \(p\leq a\) and \(F^pL_\lambda=F^pL|U_\lambda\) for \(p>a\); this is a filtration, because the kernel of \(L_\lambda\to E_\lambda\) is \(F^{a+1}L|U_\lambda\). Then \(\operatorname{gr}^aL_\lambda=E_\lambda\) and \(\operatorname{gr}^pL_\lambda=\operatorname{gr}^pL|U_\lambda\) for \(p\neq a\). The maps \(L_\lambda\to L|U_\lambda\) are filtered, and, as in the proof of Lemma 4.1 of the lesson on non-additive functors, pulling back the exact sequence \(0\to F^{a+1}L\to L\to E\to0\) along the \(E_\lambda\) and using the exactness of local inductive limits shows that \(L\) is the local inductive limit of the \(L_\lambda\) as a filtered module. By Lemma 1.2 (7), because \(\mathbf T^n\) and \(\operatorname{gr}\) commute with local inductive limits, and by the naturality of \(c\), \(c_L\) is the local inductive limit of the \(c_{L_\lambda}\); local inductive limits of isomorphisms are isomorphisms. It therefore suffices to show that every \(c_{L_\lambda}\) is an isomorphism, which is a local property on \(U_\lambda\).

Since \(E_\lambda\) is finite free, the surjection \(L_\lambda\to E_\lambda\) has a section \(s\) on the members of a covering of \(U_\lambda\); restrict to one of them. Give \(E_\lambda\) the filtration in \([a,a]\). Then \(s\) is filtered, and \((x,e)\mapsto x+s(e)\) is an isomorphism of filtered modules \(L'\oplus E_\lambda\to L_\lambda\): its inverse \(y\mapsto(y-s(\pi y),\pi y)\), where \(\pi\colon L_\lambda\to E_\lambda\), maps \(F^pL_\lambda=F^pL\) into \(F^pL\oplus0\) for \(p>a\). By Proposition 1.5 (3), \(c_{L'\oplus E_\lambda}\) is an isomorphism as soon as \(\mu\circ(c_{L'}\otimes c_{E_\lambda})\) is. Here \(c_{E_\lambda}\) is the identity (case \(a=b\)), and \(c_{L'}\) is an isomorphism by induction. In each degree \(m\), \(\operatorname{gr}\mathbf T^m(L')\cong\mathbf T^m(\operatorname{gr}L')\) is flat, by (0.1) and because \(S\), \(\Lambda\) and \(\Gamma\) carry flat modules to flat modules (Example 4.7 of the lesson on non-additive functors). So \(\mu\colon\) \(\operatorname{gr}\mathbf T^m(L')\otimes\operatorname{gr}\mathbf T^{m'}(E_\lambda)\to\) \(\operatorname{gr}(\mathbf T^m(L')\otimes\mathbf T^{m'}(E_\lambda))\) is an isomorphism by Lemma 1.4 of the lesson on filtered tensor products. This completes the induction.

Finally \(\operatorname{gr}^k\mathbf T^n(L)\cong k^k\mathbf T^n(\operatorname{gr}L)\) is a finite sum of tensor products of modules \(\mathbf T^{\nu_i}(\operatorname{gr}^iL)\), which are flat; \(F^k\mathbf T^n(L)\) and \(\mathbf T^n(L)\) are finite iterated extensions of these, hence flat [Tag 03EY]. \(\square\)

Without flatness \(c_L\) need not be injective (Exercise 3.1).

**Proposition 1.7 (two-step filtrations).** Let \(0\to L'\to L\to L''\to0\) be an exact sequence of modules, and filter \(L\) by \(F^0L=L\), \(F^1L=L'\), \(F^2L=0\), so that \(\operatorname{gr}^0L=L''\) and \(\operatorname{gr}^1L=L'\). Let \(n\geq1\).

1. For \(0\leq k\leq n\), \(F^k\mathbf T^n(L)=\) \(\sum_{j=k}^n\operatorname{Im}\bigl(\mathbf T^j(L')\otimes\mathbf T^{n-j}(L)\to\) \(\mathbf T^n(L)\bigr)\). For \(S\) and \(\Lambda\) this is the image of \(\mathbf T^k(L')\otimes\mathbf T^{n-k}(L)\).
2. \(\mathbf T^n(L)\to\mathbf T^n(L'')\) is surjective with kernel \(F^1\mathbf T^n(L)\).
3. If \(L'\) and \(L''\) are flat, then \(c_L\) identifies \(\mathbf T^{n-p}(L'')\otimes\mathbf T^p(L')\) with \(\operatorname{gr}^p\mathbf T^n(L)\) for \(0\leq p\leq n\), and \(\mathbf T^n(L')\to\mathbf T^n(L)\) is injective with image \(F^n\mathbf T^n(L)\).

For \(\Gamma\), the single term \(\operatorname{Im}(\Gamma^k(L')\otimes\Gamma^{n-k}(L))\) can be strictly smaller than \(F^k\Gamma^n(L)\), and the image of \(L'\otimes\Gamma^{n-1}(L)\) strictly smaller than the kernel in (2) (Exercise 3.2).

**Proof.** (1) The multi-indices are \(\nu=(n-j)\delta_0+j\delta_1\), of weight \(j\), and the image of \(m_\nu\) is that of \(\mathbf T^{n-j}(L)\otimes\mathbf T^j(L')\), the order of the factors not mattering for images. For \(S\) and \(\Lambda\) and \(j\geq k\), \(\mathbf T^j(L')\) is the image of \(\mathbf T^k(L')\otimes\mathbf T^{j-k}(L')\) under the product, and \(\mathbf T^{j-k}(L')\) maps into \(\mathbf T^{j-k}(L)\); so the term of index \(j\) lies in the term of index \(k\).

(2) Lemma 1.3 with \(N=L'\), and (1) for \(k=1\).

(3) \(\operatorname{gr}L=L''\oplus L'\), in weights \(0\) and \(1\), is flat, so \(c_L\) is an isomorphism (Proposition 1.6), and by (0.1) the part of weight \(p\) of \(\mathbf T^n(\operatorname{gr}L)\) is \(\mathbf T^{n-p}(L'')\otimes\mathbf T^p(L')\). For \(p=n\), \(\operatorname{gr}^n\mathbf T^n(L)=F^n\mathbf T^n(L)\) because the filtration lies in \([0,n]\), and \(c_{n\delta_1}\colon\mathbf T^n(L')\to F^n\mathbf T^n(L)\) is the map induced by \(\mathbf T^n(L'\to L)\). \(\square\)

## 2. Powers of filtered and graded complexes

For a complex \(X\), \(\mathbf T^n(X)=\operatorname{Tot}N\mathbf T^n(\Gamma\,iX)\), with \(\mathbf T^n\) applied to the terms of the mixed object \(\Gamma\,iX\) (Section 2.2 of the lesson on non-additive functors). Each term of \(\Gamma\,iX\) is a finite direct sum of terms of \(X\), and each structure map is a matrix whose entries are, up to sign, identity maps, differentials of \(X\) or zero. So for \(L\in CF(A)\) with filtration in \([a,b]\), filtering every term by the direct sums of the filtrations of the terms of \(L\) makes \(\Gamma\,iL\) a mixed object of filtered modules with filtrations in \([a,b]\), and \(\operatorname{gr}(\Gamma\,iL)=\Gamma\,i(\operatorname{gr}L)\). Likewise \(\Gamma\,iE\) is a mixed object of graded modules for \(E\in CG(A)\).

**Definition 2.1.** For \(L\in CF(A)\) and \(n\geq0\), put \(F^k\mathbf T^n(L)=\operatorname{Tot}N\,F^k\mathbf T^n(\Gamma\,iL)\), the functor \(F^k\mathbf T^n\) of Definition 1.1 being applied termwise (it is a functor by Lemma 1.2 (2)). For \(E\in CG(A)\), put \(k^p\mathbf T^n(E)=\operatorname{Tot}N\,k^p\mathbf T^n(\Gamma\,iE)\).

Since \(N\) and \(\operatorname{Tot}\) are exact, the \(F^k\mathbf T^n(L)\) are subcomplexes of \(\mathbf T^n(L)\), \(\operatorname{gr}^k\mathbf T^n(L)=\operatorname{Tot}N\operatorname{gr}^k\mathbf T^n(\Gamma\,iL)\), and \(\mathbf T^n(L)\in CF(A)\) has filtration in \([na,nb]\) by Lemma 1.2 (1); likewise \(\mathbf T^n(E)=\bigoplus_pk^p\mathbf T^n(E)\in CG(A)\). This defines functors \(\mathbf T^n\colon CF(A)\to CF(A)\) and \(CG(A)\to CG(A)\).

**Proposition 2.2.** Let \(L\in CF(A)\), \(E\in CG(A)\) with components \(E_i=k^iE\), and \(n\geq0\).

1. Applied termwise, Construction 1.4 gives a morphism of graded complexes \(c_L\colon\mathbf T^n(\operatorname{gr}L)\to\operatorname{gr}\mathbf T^n(L)\), natural in \(L\). It is an isomorphism when \(L\) has flat graded pieces.
2. \(\mathbf T^n(E)=\) \(\bigoplus_{|\nu|=n}\operatorname{Tot}N\bigl(\bigotimes_i\mathbf T^{\nu_i}(\Gamma\,iE_i)\bigr)\), where \(k^p\mathbf T^n(E)\) is the sum of the summands with \(w(\nu)=p\), and the iterated map (5.1) of the lesson on non-additive functors gives natural homotopy equivalences
\[
\begin{aligned}&\bigotimes_i\mathbf T^{\nu_i}(E_i)\\&\quad \longrightarrow\operatorname{Tot}N\Bigl(\bigotimes_i\mathbf T^{\nu_i}(\Gamma\,iE_i)\Bigr),\end{aligned}
\]
the tensor products being taken in increasing order of \(i\); factors with \(\nu_i=0\) are \(A\).
3. \(\mathbf T^n(L(m))=\mathbf T^n(L)(nm)\), \(\mathbf T^n(E(m))=\mathbf T^n(E)(nm)\), and \(\mathbf T^n(E^{\mathrm{fil}})=\mathbf T^n(E)^{\mathrm{fil}}\), where \(E^{\mathrm{fil}}\) is the filtered complex with \(F^kE^{\mathrm{fil}}=\bigoplus_{i\geq k}E_i\).
4. If \(L^q=0\) for \(q>c\), then \(\mathbf T^n(L)^q=0\) for \(q>c\). If moreover \(L\) has flat graded pieces, so has \(\mathbf T^n(L)\).

**Proof.** (1) By Proposition 1.5 (1), the maps \(c\) of the terms commute with the structure maps, so they form a morphism of mixed objects \(\mathbf T^n(\Gamma\,i\operatorname{gr}L)=\) \(\mathbf T^n(\operatorname{gr}\Gamma\,iL)\to\) \(\operatorname{gr}\mathbf T^n(\Gamma\,iL)\); apply \(\operatorname{Tot}N\), which commutes with \(\operatorname{gr}\). If \(L\) has flat graded pieces, every term of \(\operatorname{gr}\Gamma\,iL\) is a finite direct sum of modules \(\operatorname{gr}^pL^q\), hence flat, and Proposition 1.6 applies to every term.

(2) \(\Gamma\,iE=\bigoplus_i\Gamma\,iE_i\), and (0.1), being natural, is an isomorphism of mixed objects \(\bigoplus_\nu\bigotimes_i\mathbf T^{\nu_i}(\Gamma\,iE_i)\to\mathbf T^n(\Gamma\,iE)\); the weights are those of Definition 1.1. The map (5.1) is a natural homotopy equivalence and is associative, so its iterates are natural homotopy equivalences.

(3) Lemma 1.2 (4) and (6), termwise.

(4) Proposition 2.2 of the lesson on non-additive functors. For flatness, \(\operatorname{gr}\mathbf T^n(L)\cong\mathbf T^n(\operatorname{gr}L)\) by (1), and \(\operatorname{gr}L\) is a bounded above complex of flat modules. As in the proof of Proposition 4.6 there, every term of \(\mathbf T^n(X)\), for such a complex \(X\), is a finite direct sum of cross-effects of \(\mathbf T^n\) evaluated on finite direct sums of terms of \(X\), and each cross-effect is a direct summand of a value of \(\mathbf T^n\) on a flat module, which is flat (Example 4.7 there). \(\square\)

**Lemma 2.3.** Let \(f\colon L\to M\) be a filtered quasi-isomorphism between bounded above filtered complexes with flat graded pieces. Then \(\mathbf T^n(f)\) is a filtered quasi-isomorphism for every \(n\). The same holds for graded complexes.

**Proof.** By Proposition 2.2 (1), \(\operatorname{gr}\mathbf T^n(f)\) is identified with \(\mathbf T^n(\operatorname{gr}f)\). Here \(\operatorname{gr}f=\bigoplus_p\operatorname{gr}^pf\) is a quasi-isomorphism between bounded above complexes of flat modules, so \(\mathbf T^n(\operatorname{gr}f)\) is a quasi-isomorphism (Lemma 4.1 of the lesson on non-additive functors). It preserves the decomposition by weight, so every \(k^p\mathbf T^n(\operatorname{gr}f)\cong\operatorname{gr}^p\mathbf T^n(f)\) is a quasi-isomorphism, and Lemma 2.1 of the lesson on filtered derived categories applies. In the graded case apply Lemma 4.1 there directly to \(f\), and use the decomposition by weight. \(\square\)

**Proposition 2.4.** Let \(M\in CF(A)\) have filtration in \([a,b]\), with \(M^q=0\) for \(q>c\). There is a filtered quasi-isomorphism \(P\to M\), surjective in each degree, with \(P^q=0\) for \(q>c\), filtration in \([a,b]\) and flat graded pieces. The same holds for graded complexes.

**Proof.** Repeat the proof of Proposition 2.3 (1) of the lesson on filtered tensor products, replacing the K-flat resolutions [Tag 06YS] by the resolutions of [Tag 05T7] (1), applied with the flat modules, every module being a quotient of a flat module [Tag 03EW]. With \(M_p=M/F^pM\) and \(P_p\) constructed by induction, the fibre product \(X=P_p\times_{M_p}M_{p+1}\) vanishes in degrees \(>c\), so [Tag 05T7] (1) gives a quasi-isomorphism \(P_{p+1}\to X\), surjective in each degree, from a complex of flat modules vanishing in degrees \(>c\). The kernels \(G_p\) of \(P_{p+1}\to P_p\) have flat terms [Tag 03EY], and the rest of the argument is unchanged; the resulting \(P\) has \(\operatorname{gr}^pP\cong G_p\). For graded complexes apply [Tag 05T7] (1) to each component. \(\square\)

**Proposition 2.5.** Let \(KF^-(A)\subset KF(A)\) be the full subcategory of bounded above filtered complexes, and \(KF^-_{\mathrm{fl}}(A)\subset KF^-(A)\) that of the bounded above filtered complexes with flat graded pieces. Both are triangulated, and the functor \(KF^-_{\mathrm{fl}}(A)[FQis^{-1}]\to DF(A)\) is fully faithful with essential image \(D^-F(A)\). The same holds for graded complexes.

**Proof.** Cones of morphisms of bounded above complexes are bounded above, and \(\operatorname{gr}C(f)=C(\operatorname{gr}f)\) has flat terms when the terms of \(\operatorname{gr}\) of source and target are flat; so both subcategories are triangulated, and the filtered quasi-isomorphisms between their objects form multiplicative systems, the constructions of the axioms by cones staying in the subcategories. Let \(L\in KF^-(A)\), vanishing in degrees \(>c\), and let \(s\colon L'\to L\) be a filtered quasi-isomorphism with \(L'\in KF(A)\). Then \(H^q(\operatorname{gr}^pL')=0\) for \(q>c\), so \(\tau_{\leq c}L'\to L'\) is a filtered quasi-isomorphism (Lemma 3.1 of the lesson on filtered derived categories), and \(s\circ(\tau_{\leq c}L'\to L')\) has its source in \(KF^-(A)\). So the filtered quasi-isomorphisms with source in \(KF^-(A)\) and target \(L\) are cofinal among all of them, and the colimits computing morphisms out of \(L\) in \(KF^-(A)[FQis^{-1}]\) and in \(DF(A)\) agree [Tag 05Q4], as in the proof of Proposition 3.3 (a) of that lesson. So \(KF^-(A)[FQis^{-1}]\to DF(A)\) is fully faithful, and its essential image is \(D^-F(A)\) by Lemma 3.1 there. By Proposition 2.4 every object of \(KF^-(A)\) is the target of a filtered quasi-isomorphism from \(KF^-_{\mathrm{fl}}(A)\), and [Tag 0GSL] identifies \(KF^-_{\mathrm{fl}}(A)[FQis^{-1}]\) with \(KF^-(A)[FQis^{-1}]\). \(\square\)

**Theorem 2.6.** For every \(n\geq0\), \(\mathbf T^n\) induces functors
\[
\begin{gathered}L\mathbf T^n\colon D^-F(A)\to D^-F(A),\\ L\mathbf T^n\colon D^-G(A)\to D^-G(A),\end{gathered}
\]
with natural isomorphisms \(L\mathbf T^n(L)\cong\mathbf T^n(P)\) for every filtered (resp. graded) quasi-isomorphism \(P\to L\) with \(P\) bounded above with flat graded pieces. Moreover, for \(L\in D^-F(A)\):

1. The underlying object of \(L\mathbf T^n(L)\) is \(L\mathbf T^n\) of the underlying object of \(L\), with \(L\mathbf T^n\colon D^-(A)\to D^-(A)\) as in Theorem 4.4 of the lesson on non-additive functors. For \(K\in D^-(A)\) with filtration in \([0,0]\), \(L\mathbf T^n(K)\) is \(L\mathbf T^n(K)\) with filtration in \([0,0]\).
2. \(c\) induces a natural isomorphism \(L\mathbf T^n(\operatorname{gr}L)\cong\operatorname{gr}L\mathbf T^n(L)\) in \(D^-G(A)\).
3. For every \(p\) there are natural isomorphisms in \(D^-(A)\)
\[
\begin{aligned}&\bigoplus_{\substack{|\nu|=n\\ w(\nu)=p}}\ \bigotimes^{\mathbf L}_{i}L\mathbf T^{\nu_i}(\operatorname{gr}^iL)\\&\quad \xrightarrow{\ \sim\ }\operatorname{gr}^pL\mathbf T^n(L),\end{aligned}
\]
the derived tensor products being taken in increasing order of \(i\), the factors with \(\nu_i=0\) being \(A\). For \(E\in D^-G(A)\), likewise \(k^pL\mathbf T^n(E)\cong\) \(\bigoplus_{|\nu|=n,\,w(\nu)=p}\bigotimes^{\mathbf L}_iL\mathbf T^{\nu_i}(k^iE)\).
4. \(L\mathbf T^n(L(m))\cong L\mathbf T^n(L)(nm)\), and if \(\operatorname{gr}^pL\cong0\) for \(p\notin[a,b]\), then \(\operatorname{gr}^qL\mathbf T^n(L)\cong0\) for \(q\notin[na,nb]\).

**Proof.** Let \(CF^-_{\mathrm{fl}}(A)\) be the category of bounded above filtered complexes with flat graded pieces and filtered morphisms. By Proposition 2.2 (4), \(\mathbf T^n\) maps it into itself, and by Lemma 2.3 it maps filtered quasi-isomorphisms to filtered quasi-isomorphisms. Let \(f,g\colon P\to Q\) be filtered homotopic morphisms of \(CF^-_{\mathrm{fl}}(A)\). With the cylinder \(\operatorname{Cyl}(P)\) and the maps \(i_0,i_1,\pi\) and \(H\) of the proof of Proposition 3.3 (b) of the lesson on filtered derived categories, \(\operatorname{Cyl}(P)\in CF^-_{\mathrm{fl}}(A)\), \(\pi\) is a filtered quasi-isomorphism with \(\pi i_0=\pi i_1=1\), and \(f=Hi_0\), \(g=Hi_1\). In \(DF(A)\), \(\mathbf T^n(\pi)\) is invertible, so \(\mathbf T^n(i_0)=\mathbf T^n(i_1)\), and \(\mathbf T^n(f)=\mathbf T^n(g)\). Hence \(\mathbf T^n\) induces a functor \(KF^-_{\mathrm{fl}}(A)\to DF(A)\) that sends filtered quasi-isomorphisms to isomorphisms, and so a functor on \(KF^-_{\mathrm{fl}}(A)[FQis^{-1}]\). Composed with a quasi-inverse of the equivalence of Proposition 2.5, this is \(L\mathbf T^n\); its values lie in \(D^-F(A)\) by Proposition 2.2 (4), and \(L\mathbf T^n(L)\cong\mathbf T^n(P)\) for every such resolution \(P\to L\). The graded case is the same.

(1) For a resolution \(P\to L\), the underlying morphism is a quasi-isomorphism from a bounded above complex with flat terms (Definition 1.3 of the lesson on filtered tensor products), and the underlying complex of \(\mathbf T^n(P)\) is \(\mathbf T^n\) of the underlying complex of \(P\). If \(P\) has filtration in \([0,0]\), so has \(\mathbf T^n(P)\) (Lemma 1.2 (1)).

(2) \(\operatorname{gr}P\to\operatorname{gr}L\) is a graded flat resolution, and \(c_P\) is an isomorphism (Proposition 2.2 (1)); naturality comes from that of \(c\).

(3) By (2) and Proposition 2.2 (2), \(\operatorname{gr}^pL\mathbf T^n(L)\) is represented by \(\bigoplus_\nu\bigotimes_i\mathbf T^{\nu_i}(\operatorname{gr}^iP)\). Each \(\mathbf T^{\nu_i}(\operatorname{gr}^iP)\) represents \(L\mathbf T^{\nu_i}(\operatorname{gr}^iL)\) and is a bounded above complex of flat modules, hence K-flat [Tag 06YQ], so the tensor product represents the derived tensor product. The graded case is the same.

(4) Proposition 2.2 (3). For the second assertion, \(L\) is represented by a filtered complex with filtration in \([a,b]\) (Proposition 3.5 (1) of the lesson on filtered derived categories), which may be taken bounded above by Lemma 3.1 there, the truncation keeping the filtration interval; Proposition 2.4 then gives a resolution \(P\) with filtration in \([a,b]\), and \(\mathbf T^n(P)\) has filtration in \([na,nb]\). \(\square\)

**Proposition 2.7 (products).** For \(L\in D^-F(A)\) and \(m,n\geq0\) there are natural morphisms in \(D^-F(A)\)
\[
L\mathbf T^m(L)\otimes^{\mathbf L}L\mathbf T^n(L)\longrightarrow L\mathbf T^{m+n}(L),
\]
associative, whose underlying morphisms are the products of Section 5 of the lesson on non-additive functors. Under the isomorphism \(\operatorname{gr}(X\otimes^{\mathbf L}Y)\cong\operatorname{gr}X\otimes^{\mathbf L}\operatorname{gr}Y\) of (3.1) in Proposition 3.2 of the lesson on filtered tensor products and Theorem 2.6 (2), the graded pieces of these morphisms are the products of \(L\mathbf T(\operatorname{gr}L)\). The same holds in \(D^-G(A)\).

**Proof.** Let \(P\to L\) be a resolution as in Theorem 2.6. The product \(\mathbf T^m(P)\otimes\mathbf T^n(P)\to\mathbf T^{m+n}(P)\) is the termwise multiplication of the mixed object \(\mathbf T(\Gamma\,iP)\) composed with (5.1). The termwise multiplication is filtered by Lemma 1.2 (3), and (5.1) is natural, so it maps \(\operatorname{Tot}NF^pX\otimes\operatorname{Tot}NF^qY\) into the image of \(\operatorname{Tot}N(F^pX\otimes F^qY)\), which lies in \(\operatorname{Tot}NF^{p+q}(X\otimes Y)\); so the product is a morphism of filtered complexes. Each \(\operatorname{gr}^r\mathbf T^m(P)\) is a bounded above complex of flat modules, hence K-flat [Tag 06YQ], so \(\mathbf T^m(P)\) is filtered K-flat (Definition 2.1 of the lesson on filtered tensor products) and \(\mathbf T^m(P)\otimes\mathbf T^n(P)\) represents \(L\mathbf T^m(L)\otimes^{\mathbf L}L\mathbf T^n(L)\) (Proposition 3.2 there). The products are natural transformations of functors on \(CF^-_{\mathrm{fl}}(A)\); by the cylinder argument of the proof of Theorem 2.6 and Lemma 2.3 they induce natural morphisms in \(D^-F(A)\). Associativity holds on complexes (Section 5 of the lesson on non-additive functors). For the graded pieces, \(c\) is multiplicative termwise (Proposition 1.5 (2)) and natural, and (5.1) is natural; on the filtered K-flat models the isomorphism (3.1) there is \(\mu\). \(\square\)

**Proposition 2.8 (amplitude).** Let \([a,b]\) be an interval containing \(0\), and let \(L\in D^-F(A)\) have tor amplitude (resp. perfect amplitude) in \([a,b]\) in the sense of Definition 1.1 of the lesson on perfect filtered complexes. Then \(L\mathbf T^n(L)\) has tor amplitude (resp. perfect amplitude) in \([na,nb]\). In particular \(L\mathbf T^n\) carries filtered perfect objects to filtered perfect objects. The same holds in \(D^-G(A)\).

**Proof.** By Theorem 2.6 (3) it suffices to show that a derived tensor product \(\bigotimes^{\mathbf L}_iL\mathbf T^{\nu_i}(K_i)\) with \(|\nu|=n\), where each \(K_i\in D(A)\) has tor amplitude (resp. perfect amplitude) in \([a,b]\), has the corresponding property in \([na,nb]\); finite direct sums preserve both properties.

Tor amplitude: by [Tag 08G1], \(K_i\) is represented by a complex \(X_i\) of flat modules with terms in degrees \([a,b]\). Then \(L\mathbf T^{\nu_i}(K_i)\cong\mathbf T^{\nu_i}(X_i)\) has flat terms in degrees \([\nu_ia,\nu_ib]\) (Proposition 2.2 and Example 4.7 of the lesson on non-additive functors; for \(\nu_i=0\) it is \(A\) in degree \(0\)), and the tensor product of these complexes is a complex of flat modules in degrees \([na,nb]\) that represents the derived tensor product [Tag 06YQ]. By [Tag 08G1] again it has tor amplitude in \([na,nb]\).

Perfect amplitude: the question is local, and \(L\mathbf T^{\nu_i}\) commutes with restriction to objects of \(T\), because \(\mathbf T^{\nu_i}\) of complexes does and restriction preserves bounded above flat resolutions. After refining a covering finitely many times, each \(K_i\) is a strictly perfect complex \(X_i\) with terms in \([a,b]\), whose terms are direct summands of finite free modules. As in the proof of Proposition 4.6 of the lesson on non-additive functors, the terms of \(\mathbf T^{\nu_i}(X_i)\) are finite direct sums of direct summands of modules \(\mathbf T^{\nu_i}(Y)\), \(Y\) a finite direct sum of terms of \(X_i\). Such a \(Y\) is a direct summand of a finite free module \(Y\oplus Y'\), and \(\mathbf T^{\nu_i}(Y)\) is a direct summand of \(\mathbf T^{\nu_i}(Y\oplus Y')\) by (0.1), which is finite free. So \(\mathbf T^{\nu_i}(X_i)\) is strictly perfect with terms in \([\nu_ia,\nu_ib]\), and the tensor product of these strictly perfect complexes is strictly perfect with terms in \([na,nb]\). \(\square\)

## 3. Exercises

**Exercise 3.1.** Let \(T\) be the punctual topos and \(A=\mathbf Z\). Let \(L=\mathbf Z\) with \(F^0L=\mathbf Z\), \(F^1L=2\mathbf Z\), \(F^2L=0\). Show that \(\Lambda^2(L)=0\), while the part of weight \(1\) of \(\Lambda^2(\operatorname{gr}L)\) is isomorphic to \(\mathbf Z/2\). Conclude that \(c_L\) is not injective, and explain why Proposition 1.6 does not apply.

**Exercise 3.2.** Let \(T\) be the punctual topos, \(A=\mathbf Z\), and \(L=L'=\mathbf Z\), \(L''=0\), filtered as in Proposition 1.7. Show that \(F^1\Gamma^2(L)=\Gamma^2(L)\), whereas the image of \(\Gamma^1(L')\otimes\Gamma^1(L)\) is \(2\,\Gamma^2(L)\). Show that the image of \(L'\otimes\Gamma^1(L)\) is not the kernel of \(\Gamma^2(L)\to\Gamma^2(L'')\), and that for \(S^2\) the corresponding statements are true.

**Exercise 3.3.** Let \(K\in D^-(A)\), and let \(L\in D^-F(A)\) be \(K\oplus K\) with \(F^0L=L\), \(F^1L=0\oplus K\), \(F^2L=0\). Compute the graded pieces of \(LS^2(L)\) and of \(L\Lambda^2(L)\).

**Exercise 3.4.** Let \(L\in D^-F(A)\) have \(\operatorname{gr}^0L\cong A\), \(\operatorname{gr}^1L\cong A[-1]\) and \(\operatorname{gr}^pL\cong0\) for \(p\neq0,1\). Show that the graded pieces of \(L\Lambda^2(L)\) are \(0\), \(A[-1]\) and \(A[-2]\) in weights \(0\), \(1\), \(2\), and check the bound of Proposition 2.8. (Use the shift isomorphisms of Theorem 3.2 of the lesson on Koszul complexes.)

## 4. Complete solutions

**Solution 3.1.** \(\Lambda^2\) of a module generated by one element is zero, since \(x\wedge x=0\) and \(\Lambda^2\) is generated by the products of generators. Here \(\operatorname{gr}^0L=\mathbf Z/2\mathbf Z\) and \(\operatorname{gr}^1L=2\mathbf Z\cong\mathbf Z\). By (0.1), \(\Lambda^2(\operatorname{gr}L)=\) \(\Lambda^2(\operatorname{gr}^0L)\oplus(\operatorname{gr}^0L\otimes\operatorname{gr}^1L)\oplus\Lambda^2(\operatorname{gr}^1L)\) in weights \(0,1,2\); the outer terms vanish, and the middle one is \(\mathbf Z/2\otimes\mathbf Z\cong\mathbf Z/2\). Its image in \(\operatorname{gr}^1\Lambda^2(L)=0\) is zero, so \(c_L\) is not injective. Proposition 1.6 does not apply because \(\operatorname{gr}^0L=\mathbf Z/2\) is not flat.

**Solution 3.2.** Write \(e=1\). \(\Gamma(\mathbf Z)\) is free on the \(\gamma_k(e)\), so \(\Gamma^2(L)=\mathbf Z\gamma_2(e)\), and \(e\cdot e=\gamma_1(e)\gamma_1(e)=2\gamma_2(e)\). In Definition 1.1 the multi-indices with \(|\nu|=2\) supported in \([0,1]\) are \(2\delta_0\), \(\delta_0+\delta_1\), \(2\delta_1\), of weights \(0,1,2\), with images \(\Gamma^2(L)\), \(\mathbf Z e\cdot e=2\Gamma^2(L)\) and \(\Gamma^2(F^1L)=\Gamma^2(L)\). So \(F^1\Gamma^2(L)=\Gamma^2(L)\), while the term \(j=k=1\) alone gives \(2\Gamma^2(L)\); the term \(j=2\) is needed. Since \(L''=0\), the kernel of \(\Gamma^2(L)\to\Gamma^2(L'')=0\) is \(\Gamma^2(L)\), but the image of \(L'\otimes\Gamma^1(L)\) is \(2\Gamma^2(L)\). For \(S^2\): \(S^2(L)=\mathbf Ze^2\) and the image of \(L'\otimes S^1(L)\) is \(\mathbf Ze\cdot e=S^2(L)\), as Proposition 1.7 (1) and (2) state.

**Solution 3.3.** \(\operatorname{gr}^0L\cong K\cong\operatorname{gr}^1L\), and the other graded pieces vanish. By Theorem 2.6 (3), with the multi-indices \(2\delta_0\), \(\delta_0+\delta_1\), \(2\delta_1\) of weights \(0,1,2\): \(\operatorname{gr}^0LS^2(L)\cong LS^2(K)\), \(\operatorname{gr}^1LS^2(L)\cong K\otimes^{\mathbf L}K\), \(\operatorname{gr}^2LS^2(L)\cong LS^2(K)\), and the others vanish. Likewise \(L\Lambda^2(K)\), \(K\otimes^{\mathbf L}K\), \(L\Lambda^2(K)\) for \(L\Lambda^2(L)\). The underlying object is, by Theorem 2.6 (1) and the exponential isomorphism, \(LS^2(K)\oplus(K\otimes^{\mathbf L}K)\oplus LS^2(K)\) (resp. with \(\Lambda^2\)), in accordance with the graded pieces.

**Solution 3.4.** By Theorem 2.6 (3) with the same three multi-indices: weight \(0\) gives \(L\Lambda^2(A)\otimes^{\mathbf L}A=\Lambda^2(A)=0\), as \(A\) is flat and generated by one element; weight \(1\) gives \(A\otimes^{\mathbf L}A[-1]=A[-1]\); weight \(2\) gives \(L\Lambda^2(A[-1])\). By Theorem 3.2 (i) of the lesson on Koszul complexes applied to \(E=A[-1]\), \(LS^2(A)\cong L\Lambda^2(A[-1])[2]\), so \(L\Lambda^2(A[-1])\cong S^2(A)[-2]=A[-2]\). The object \(L\) has perfect amplitude in \([0,1]\), an interval containing \(0\), and Proposition 2.8 gives perfect amplitude in \([0,2]\) for \(L\Lambda^2(L)\); indeed \(A[-1]\) and \(A[-2]\) are strictly perfect with terms in degrees \(1\) and \(2\).

## Sources

The filtrations of Definition 1.1, the comparison map (1.1) and Proposition 1.6, and the derived functors of Theorem 2.6 with their graded pieces, products and amplitude, are due to Illusie, in his work on the cotangent complex. Proposition 1.6 is proved there, as here, by reduction to a direct sum through the theorem of Deligne and Lazard. The two-step filtration of Proposition 1.7 is the classical Koszul filtration of a power of an extension. Bounded above flat resolutions are constructed in the AI-integrated Stacks project, Chapter *Derived Categories*.