Filtered Koszul complexes and the shift isomorphisms

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

The shift formulas $LS^n(E[1])\cong L\Lambda^n(E)[n]$ and $L\Lambda^n(E[1])\cong L\Gamma^n(E)[n]$ come from the Koszul complexes of the exact sequence $0\to E\to C(E)\to E[1]\to0$ of the cone. This lesson filters the Koszul complexes and derives the shift formulas on the bounded above filtered and graded derived categories of a ringed topos. It proves:

the Koszul complexes of a filtered map are filtered, and their graded pieces are the Koszul complexes of the graded map when the graded pieces are flat; the Koszul sequences of a sequence of filtered modules with exact flat graded pieces are exact on every filtration step;
the shift isomorphisms exist on $D^-F(A)$ and $D^-G(A)$, with the isomorphisms of complexes of modules as underlying morphisms, and they are compatible with the products, with the sign $(-1)^{mn}$ for the second one;
on the graded pieces, the shift isomorphism of a filtered object is a sum over the multi-indices $\nu$ of the tensor products of the shift isomorphisms of the graded pieces, composed with the canonical isomorphisms $\tau$ of shifted tensor products; for the second shift isomorphism each term carries the sign $(-1)^{\sum_{i<j}\nu_i\nu_j}$.

Prerequisites. Koszul complexes and the shift formulas (all of it); Filtered and graded powers of complexes (the conventions, Definition 1.1, Lemma 1.2, Propositions 1.5, 1.6, 2.2, 2.5, 2.7, Theorem 2.6); Non-additive functors of complexes in a ringed topos (Proposition 2.3, Section 5); Filtered derived categories (the conventions on cones and triangles, Definition 1.1, Lemma 2.1, Proposition 2.2); Filtered tensor products on a ringed topos (Lemmas 1.4 and 2.2). From the AI-integrated Stacks project: the triangle of a short exact sequence of complexes [Stacks, Tag 0152] flatness in short exact sequences [Tag 03EY] and bounded above complexes of flat modules are K-flat [Tag 06YQ].

Conventions

As in the lesson on filtered and graded powers of complexes: $A$ is a commutative ring of a topos $T$; filtered modules have finite filtrations; $S^n$, $\Lambda^n$, $\Gamma^n$ of a filtered module carry the filtrations of Definition 1.1 there, and $c$ is the comparison map (1.1) there; $\mu\colon\operatorname{gr}X\otimes\operatorname{gr}Y\to\operatorname{gr}(X\otimes Y)$ is the map (1.1) of the lesson on filtered tensor products; multi-indices $\nu$ have $|\nu|=\sum_i\nu_i$ and weight $w(\nu)=\sum_ii\nu_i$. Put $e(\nu)=\sum_{i<j}\nu_i\nu_j .$ The Koszul complexes $\operatorname{Kos}_\bullet(u)=(\Lambda E\otimes SF,d)$ and $\operatorname{Kos}^\bullet(u)=(\Gamma E\otimes\Lambda F,d)$ of a map $u\colon E\to F$, their parts $k^n$ of total degree $n$, and the maps of Construction 1.3 are those of the lesson on Koszul complexes. For complexes $X$, $Y$, $\tau\colon X[m]\otimes Y[n]\to(X\otimes Y)[m+n]$ is $x\otimes y\mapsto(-1)^{n|x|}x\otimes y$, $|x|$ the degree of $x$ in $X[m]$, as in Section 3 there. It is associative: for $x$, $y$, $z$ of degrees $|x|$, $|y|$, $|z|$ in $X[a]$, $Y[b]$, $Z[c]$, both $\tau\circ(\tau\otimes1)$ and $\tau\circ(1\otimes\tau)$ multiply $x\otimes y\otimes z$ by $(-1)^{(b+c)|x|+c|y|}$. For complexes $X_1,\ldots,X_r$ and integers $n_1,\ldots,n_r$, $\tau\colon$ $X_1[n_1]\otimes\cdots\otimes X_r[n_r]\to$ $(X_1\otimes\cdots\otimes X_r)[n_1+\cdots+n_r]$ denotes the iterated map. Triangles of short exact sequences follow the conventions of the lesson on filtered derived categories: $0\to X\xrightarrow fY\to Z\to0$ gives $X\to Y\to Z\xrightarrow\delta X[1]$ with $\delta=-p\circ q^{-1}$, for $q\colon C(f)\to Z$ and $p\colon C(f)\to X[1]$ the obvious maps [Tag 0152].

1. Filtered Koszul complexes

Proposition 1.1. Let $u\colon E\to F$ be a filtered map of filtered modules, and give $\Lambda^pE\otimes S^qF$ and $\Gamma^pE\otimes\Lambda^qF$ the tensor products of the filtrations of the factors.

The differentials of $\operatorname{Kos}_\bullet(u)$ and $\operatorname{Kos}^\bullet(u)$ are filtered, and so are their multiplications. For a graded map of graded modules, the Koszul complexes are complexes of graded modules, with graded differentials and multiplications.
The maps $\kappa=$ $\mu\circ(c\otimes c)\colon$ $\Lambda(\operatorname{gr}E)\otimes S(\operatorname{gr}F)\to$ $\operatorname{gr}(\Lambda E\otimes SF)$ and $\kappa=$ $\mu\circ(c\otimes c)\colon$ $\Gamma(\operatorname{gr}E)\otimes\Lambda(\operatorname{gr}F)\to$ $\operatorname{gr}(\Gamma E\otimes\Lambda F)$ are multiplicative morphisms of complexes $\operatorname{Kos}(\operatorname{gr}u)\to\operatorname{gr}\operatorname{Kos}(u)$, natural in $u$. They are isomorphisms when $\operatorname{gr}E$ and $\operatorname{gr}F$ are flat.

Proof. (1) By the remarks after Definition 1.1 of the lesson on filtered and graded powers, $F^k(\Lambda^pE\otimes S^qF)$ is generated by the local sections $x_1\wedge\cdots\wedge x_p\otimes y_1\cdots y_q$ with $x_j\in F^{i_j}E$, $y_l\in F^{t_l}F$ and $\sum_ji_j+\sum_lt_l\geq k$. The differential sends such a section to $\sum_j(-1)^{j-1}x_1\wedge\cdots\widehat{x_j}\cdots\wedge x_p\otimes$ $u(x_j)y_1\cdots y_q$, and $u(x_j)\in F^{i_j}F$; each term has the same weight. Likewise $F^k(\Gamma^pE\otimes\Lambda^qF)$ is generated by the sections $\gamma_{n_1}(x_1)\cdots\gamma_{n_r}(x_r)\otimes z_1\wedge\cdots\wedge z_q$ with $x_j\in F^{i_j}E$, $z_l\in F^{t_l}F$ and $\sum_jn_ji_j+\sum_lt_l\geq k$. The differential is a derivation, and it replaces one factor $\gamma_{n_j}(x_j)$, of weight $n_ji_j$, by $\gamma_{n_j-1}(x_j)$ times $u(x_j)\in F^{i_j}F$ in the exterior factor, of total weight $n_ji_j$; so it is filtered. The multiplications are filtered by Lemma 1.2 (3) of the lesson on filtered and graded powers. The graded case is the same with homogeneous elements.

(2) $c$ is multiplicative (Proposition 1.5 (2) of the lesson on filtered and graded powers). The multiplications of $\Lambda E\otimes SF$ and $\Gamma E\otimes\Lambda F$ are those of tensor products of algebras, without signs, since the second factor of the first algebra and the first factor of the second have exterior degree $0$; so $\mu$ is multiplicative, and so is $\kappa$. Let $D_1=\kappa\circ d$, with $d$ the differential of $\operatorname{Kos}(\operatorname{gr}u)$, and $D_2=\operatorname{gr}(d)\circ\kappa$. As $d$ and $\operatorname{gr}(d)$ are derivations (with the sign rule of the first, respectively second, degree) and $\kappa$ is multiplicative, both satisfy $D(\alpha\beta)=D(\alpha)\kappa(\beta)\pm\kappa(\alpha)D(\beta)$ with the same sign; so the local sections on which $D_1$ and $D_2$ agree form a subalgebra. They agree on generators. For $\operatorname{Kos}_\bullet$: on $\bar x\otimes1$, $\bar x\in\operatorname{gr}^iE$ the class of $x\in F^iE$, both give the class of $1\otimes u(x)$; on $1\otimes\bar y$ both vanish. For $\operatorname{Kos}^\bullet$: on $\gamma_k(\bar x)\otimes1$, both give the class of $\gamma_{k-1}(x)\otimes u(x)$, by the description of $c$ on divided powers; on $1\otimes\bar z$ both vanish. These generate $\Lambda(\operatorname{gr}E)\otimes S(\operatorname{gr}F)$, respectively $\Gamma(\operatorname{gr}E)\otimes\Lambda(\operatorname{gr}F)$. So $D_1=D_2$. Naturality comes from that of $c$ and $\mu$. If $\operatorname{gr}E$ and $\operatorname{gr}F$ are flat, the maps $c$ are isomorphisms (Proposition 1.6 there) and $\Lambda^pE$, $\Gamma^pE$ have flat graded pieces, so $\mu$ is an isomorphism (Lemma 1.4 of the lesson on filtered tensor products). $\square$

Proposition 1.2. Let $0\to E\xrightarrow{u}F\xrightarrow{v}G\to0$ be a sequence of filtered maps of filtered modules such that $0\to\operatorname{gr}E\to\operatorname{gr}F\to\operatorname{gr}G\to0$ is an exact sequence of flat modules. Then $E$, $F$, $G$ are flat, the sequence is exact on every filtration step, and for $n\geq1$ the Koszul sequences $\begin{gathered} 0\to\Lambda^nE\to\Lambda^{n-1}E\otimes F\to\cdots\\ \cdots\to E\otimes S^{n-1}F\to S^nF\to S^nG\to0,\\[4pt] 0\to\Gamma^nE\to\Gamma^nF\to\Gamma^{n-1}F\otimes G\to\cdots\\ \cdots\to F\otimes\Lambda^{n-1}G\to\Lambda^nG\to0 \end{gathered}$ are sequences of filtered maps. Through $\kappa$ and $c$, their graded pieces are the Koszul sequences of $0\to\operatorname{gr}E\to\operatorname{gr}F\to\operatorname{gr}G\to0$; in particular they are exact on every graded piece and on every filtration step.

Proof. Modules with a finite filtration whose graded pieces are flat are flat [Tag 03EY]. A sequence of filtered maps of modules with finite filtrations whose graded pieces form exact sequences is exact on every $F^k$: by descending induction on $k$ from $F^k=0$, with the snake lemma applied to $0\to F^{k+1}\to F^k\to\operatorname{gr}^k\to0$. The maps of Construction 1.3 of the lesson on Koszul complexes are filtered, since $S(v)$ and $\Gamma(u)$ are (Lemma 1.2 (2) of the lesson on filtered and graded powers). By Proposition 1.1 the graded pieces of the Koszul complexes are, through $\kappa$, the Koszul complexes of $\operatorname{gr}u$ and $\operatorname{gr}v$; the augmentations correspond through $c$, because $c$ and $\kappa$ are multiplicative and natural and the augmentations are homomorphisms of algebras determined by their values on generators. The Koszul sequences of the exact sequence of flat modules $0\to\operatorname{gr}E\to\operatorname{gr}F\to\operatorname{gr}G\to0$ are exact by Theorem 2.1 of the lesson on Koszul complexes. $\square$

2. The shift isomorphisms

Lemma 2.1.

Let $0\to X\xrightarrow fY\xrightarrow gZ\to0$ be a sequence of morphisms of filtered complexes whose graded pieces are exact sequences of complexes. Then $q\colon C(f)\to Z$ is a filtered quasi-isomorphism, and $X\to Y\to Z\xrightarrow\delta X[1]$, $\delta=-p\circ q^{-1}$, is a distinguished triangle of $DF(A)$. Its graded pieces are the triangles of the exact sequences of graded pieces.
Let $0\to$ $P_n\to$ $I_{n-1}\to$ $\cdots\to$ $I_0\to$ $Q\to$ $0$ be a sequence of morphisms of filtered complexes whose graded pieces are exact, with every $I_j$ filtered acyclic. Then the composite of the morphisms $\delta$ of (1) for the short sequences into which it splits is an isomorphism $Q\cong P_n[n]$ in $DF(A)$, natural for morphisms of such sequences. Its graded pieces are the isomorphisms of Lemma 3.1 of the lesson on Koszul complexes for the sequences of graded pieces.

Proof. (1) The triangle $X\to Y\to C(f)\to X[1]$ is distinguished in $KF(A)$ (Proposition 2.2 (1) of the lesson on filtered derived categories), and $\operatorname{gr}^pq\colon C(\operatorname{gr}^pf)\to\operatorname{gr}^pZ$ is a quasi-isomorphism, $0\to\operatorname{gr}^pX\to\operatorname{gr}^pY\to\operatorname{gr}^pZ\to0$ being exact [Tag 0152]; by Lemma 2.1 there $q$ is a filtered quasi-isomorphism. Replacing $C(f)$ by $Z$ through $q$ gives the triangle with $\delta$. The functors $\operatorname{gr}^p$ are exact (Proposition 2.2 (4) there) and commute with cones.

(2) By the proof of Proposition 1.2, the sequence is exact on every $F^k$. Let $K_j$ be the image of $I_j\to I_{j-1}$ for $1\leq j\leq n-1$, with $K_0=Q$, $K_n=P_n$, filtered by $F^kK_j=K_j\cap F^kI_{j-1}$. Exactness on every $F^k$ gives exact sequences $0\to F^kK_{j+1}\to F^kI_j\to F^kK_j\to0$, and by the snake lemma exact sequences of graded pieces $0\to\operatorname{gr}^kK_{j+1}\to\operatorname{gr}^kI_j\to\operatorname{gr}^kK_j\to0$. By (1) each gives a triangle $K_{j+1}\to I_j\to K_j\xrightarrow{\delta_j}K_{j+1}[1]$; since $I_j$ is filtered acyclic, $\delta_j$ is an isomorphism. The composite $\delta_{n-1}[n-1]\circ\cdots\circ\delta_0\colon Q\to P_n[n]$ is natural because the triangles of (1) are natural for morphisms of short exact sequences. Its graded pieces are the composites for the sequences of graded pieces, by the last assertion of (1). $\square$

Theorem 2.2. Let $n\geq1$. For $M\in D^-F(A)$ there are isomorphisms in $D^-F(A)$, natural in $M$, $\begin{gathered}\sigma^S_M \colon LS^n(M[1])\\ \xrightarrow{\ \sim\ }L\Lambda^n(M)[n],\\ \sigma^\Lambda_M \colon L\Lambda^n(M[1])\\ \xrightarrow{\ \sim\ }L\Gamma^n(M)[n],\end{gathered}$ whose underlying morphisms of $D^-(A)$ are the isomorphisms of Theorem 3.2 of the lesson on Koszul complexes. The same holds in $D^-G(A)$.

Proof. Let $P\to M$ be a filtered quasi-isomorphism from a bounded above filtered complex with flat graded pieces (Theorem 2.6 of the lesson on filtered and graded powers). The cone $C(P)$ of $1_P$, filtered by $F^kC(P)=F^kP\oplus F^kP[1]$, sits in the sequence $0\to P\to C(P)\to P[1]\to0$, split in each degree as a sequence of filtered modules (conventions on cones of the lesson on filtered derived categories). Applying the additive functor $\Gamma\,i$ gives a sequence of mixed objects of filtered modules, split in each term; its graded pieces are split exact sequences of flat modules. By Proposition 1.2 applied termwise, and since $N$ and $\operatorname{Tot}$ are exact, the first Koszul sequence gives a sequence of filtered complexes $\begin{gathered} 0\to\Lambda^n(P)\to I_{n-1}\to\cdots\\ \cdots\to I_0\to S^n(P[1])\to0,\\ I_{n-p}=\operatorname{Tot}N\bigl(\Lambda^{n-p}(\Gamma\,iP)\\\otimes S^p(\Gamma\,iC(P))\bigr)\ (1\leq p\leq n), \end{gathered}$ whose graded pieces are exact. Through $\kappa$, $\operatorname{gr}I_{n-p}$ is $\begin{gathered}\operatorname{Tot}N\bigl(\Lambda^{n-p}(\Gamma\,i\operatorname{gr}P)\\ \otimes S^p(\Gamma\,iC(\operatorname{gr}P))\bigr),\end{gathered}$ using $\operatorname{gr}C(P)=C(\operatorname{gr}P)$. By (5.1) of the lesson on non-additive functors this is homotopy equivalent to $\Lambda^{n-p}(\operatorname{gr}P)\otimes S^p(C(\operatorname{gr}P))$. The cone $C(\operatorname{gr}P)$ is homotopy equivalent to $0$, so $S^p(C(\operatorname{gr}P))$ is homotopy equivalent to $S^p(0)=0$ for $p\geq1$ (Proposition 2.3 there), and $I_{n-p}$ is filtered acyclic. Lemma 2.1 (2) gives an isomorphism $S^n(P[1])\cong\Lambda^n(P)[n]$ in $DF(A)$. Since $P[1]\to M[1]$ is again such a resolution, this is an isomorphism $\sigma^S_M\colon LS^n(M[1])\to L\Lambda^n(M)[n]$.

A filtered morphism $P\to P'$ induces a morphism of the sequences, so the isomorphism is natural in $P$ by Lemma 2.1 (2); by the equivalence of Proposition 2.5 of the lesson on filtered and graded powers it is natural in $M$ and independent of $P$. The underlying sequence of complexes is the one used in the proof of Theorem 3.2 of the lesson on Koszul complexes, and Lemma 2.1 (2) forgets to Lemma 3.1 there. For $\sigma^\Lambda$ use the second Koszul sequence, whose intermediate terms are $\operatorname{Tot}N\bigl(\Gamma^{n-p}(\Gamma\,iC(P))\otimes\Lambda^p(\Gamma\,i(P[1]))\bigr)$ for $0\leq p\leq n-1$; their graded pieces are acyclic because $\Gamma^{n-p}(C(\operatorname{gr}P))$ is homotopy equivalent to $0$ for $n-p\geq1$. The graded case is the same. $\square$

Proposition 2.3 (graded pieces). Let $n\geq1$.

For $M\in D^-F(A)$, under the isomorphisms $\operatorname{gr}LS^n(M[1])\cong LS^n(\operatorname{gr}M[1])$ and $\operatorname{gr}L\Lambda^n(M)\cong L\Lambda^n(\operatorname{gr}M)$ of Theorem 2.6 (2) of the lesson on filtered and graded powers, $\operatorname{gr}\sigma^S_M=\sigma^S_{\operatorname{gr}M}$, the isomorphism of Theorem 2.2 in $D^-G(A)$; likewise $\operatorname{gr}\sigma^\Lambda_M=\sigma^\Lambda_{\operatorname{gr}M}$.
Let $E\in D^-G(A)$ have components $E_i$. In the decompositions of Theorem 2.6 (3) of that lesson, $\sigma^S_E$ maps the summand $\bigotimes^{\mathbf L}_iLS^{\nu_i}(E_i[1])$ to the summand $\bigl(\bigotimes^{\mathbf L}_iL\Lambda^{\nu_i}(E_i)\bigr)[n]$, by $\tau\circ\Bigl(\bigotimes_i\sigma^S_{E_i}\Bigr),$ where $\sigma^S_{E_i}\colon LS^{\nu_i}(E_i[1])\to L\Lambda^{\nu_i}(E_i)[\nu_i]$ is the isomorphism of Theorem 3.2 of the lesson on Koszul complexes (the identity of $A$ when $\nu_i=0$) and $\tau\colon$ $\bigotimes_i\bigl(L\Lambda^{\nu_i}(E_i)[\nu_i]\bigr)\to$ $\bigl(\bigotimes_iL\Lambda^{\nu_i}(E_i)\bigr)[n]$. Likewise $\sigma^\Lambda_E$ maps $\bigotimes^{\mathbf L}_iL\Lambda^{\nu_i}(E_i[1])$ to $\bigl(\bigotimes^{\mathbf L}_iL\Gamma^{\nu_i}(E_i)\bigr)[n]$ by $(-1)^{e(\nu)}\,\tau\circ\Bigl(\bigotimes_i\sigma^\Lambda_{E_i}\Bigr).$

Consequently, for $M\in D^-F(A)$ and every $p$, $\operatorname{gr}^p\sigma^S_M$ is the sum over $|\nu|=n$, $w(\nu)=p$, of the maps $\tau\circ\bigotimes_i\sigma^S_{\operatorname{gr}^iM}$, and $\operatorname{gr}^p\sigma^\Lambda_M$ the sum of the maps $(-1)^{e(\nu)}\tau\circ\bigotimes_i\sigma^\Lambda_{\operatorname{gr}^iM}$.

Proof. (1) By Lemma 2.1 (2), $\operatorname{gr}\sigma^S_M$ is the isomorphism defined by the sequence of graded pieces of the sequence in the proof of Theorem 2.2. Through $\kappa$ and $c$ (Propositions 1.1 and 1.2, applied termwise to the mixed objects), that sequence is the sequence for the graded complex $\operatorname{gr}P$, whose isomorphism is $\sigma^S_{\operatorname{gr}M}$. The same argument applies to $\sigma^\Lambda$.

(2) It suffices to compare underlying morphisms in $D^-(A)$, the decompositions being by weight and $\sigma^S_E$ being a morphism of $D^-G(A)$. Write $E=\bigoplus_iE_i$ and $\iota_i\colon E_i\to E$ for the inclusions. On complexes, the isomorphism of Theorem 2.6 (3) of the lesson on filtered and graded powers is the iterated map (5.1) followed by the termwise exponential isomorphism (0.1) there; by the naturality of (5.1) and the definition of the products in Section 5 of the lesson on non-additive functors, it is the iterated product, in increasing order of $i$, of the maps $LS^{\nu_i}(\iota_i[1])\colon LS^{\nu_i}(E_i[1])\to LS^{\nu_i}(E[1])$, and likewise for $L\Lambda$. By naturality of the shift isomorphisms, $\sigma^S_E\circ LS^{\nu_i}(\iota_i[1])=L\Lambda^{\nu_i}(\iota_i)[\nu_i]\circ\sigma^S_{E_i}$ in degree $\nu_i$. Let $i_1<\cdots<i_r$ be the indices with $\nu_i>0$. Proposition 3.4 of the lesson on Koszul complexes, applied to the product of the factors of indices $i_1,\ldots,i_s$ (of degree $a=\nu_{i_1}+\cdots+\nu_{i_s}$) and the factor of index $i_{s+1}$ (of degree $b=\nu_{i_{s+1}}$), introduces one map $\tau$ and no sign. Induction on $s$, with the associativity of the products and of $\tau$ and the naturality of $\tau$, gives the iterated $\tau$. For $\sigma^\Lambda_E$, the second part of Proposition 3.4 there introduces at each step the sign $(-1)^{ab}=\prod_{l\leq s}(-1)^{\nu_{i_l}\nu_{i_{s+1}}}$, and the total sign is $\prod_{l<m}(-1)^{\nu_{i_l}\nu_{i_m}}=(-1)^{e(\nu)}$.

The final statement combines (1), (2) and Theorem 2.6 (2), (3) of the lesson on filtered and graded powers. $\square$

Proposition 2.4 (products). Let $M\in D^-F(A)$ and $m,n\geq1$, and let the products be those of Proposition 2.7 of the lesson on filtered and graded powers, $LS^m(M[1])\otimes^{\mathbf L}LS^n(M[1])\to$ $LS^{m+n}(M[1])$ and $L\Lambda^m(M)\otimes^{\mathbf L}L\Lambda^n(M)\to$ $L\Lambda^{m+n}(M)$. Then $\begin{gathered}\sigma^S_M\circ(\text{product})\\ =(\text{product})[m+n]\circ\tau\circ(\sigma^S_M\otimes\sigma^S_M)\end{gathered}$ as maps $LS^m(M[1])\otimes^{\mathbf L}LS^n(M[1])\to$ $L\Lambda^{m+n}(M)[m+n]$. For the second shift isomorphism, $\sigma^\Lambda_M\circ$ $(\text{product})=$ $(-1)^{mn}(\text{product})[m+n]\circ$ $\tau\circ$ $(\sigma^\Lambda_M\otimes\sigma^\Lambda_M)$, the products being those of $L\Lambda(M[1])$ and $L\Gamma(M)$. The same holds in $D^-G(A)$.

Proof. Lemmas 3.1 (2), (3) and 3.3 of the lesson on Koszul complexes hold for sequences of filtered complexes as in Lemma 2.1 (2) whose terms are bounded above with flat graded pieces, the isomorphisms being those of Lemma 2.1 (2) in $DF(A)$. Their proofs use the triangles of short exact sequences, which Lemma 2.1 (1) provides in $DF(A)$, and morphisms of complexes and homotopies built from the maps of the sequences, identities and signs, which are filtered. The quasi-isomorphisms there are filtered quasi-isomorphisms, because the graded pieces of the sequences are exact and finite sums of tensor products of bounded above complexes with flat graded pieces are filtered K-flat (Lemma 2.2 of the lesson on filtered tensor products, with [Tag 06YQ]). With $P\to M$ as in the proof of Theorem 2.2, the multiplication of the filtered Koszul algebras (Proposition 1.1), composed with (5.1), which is natural and therefore filtered, is a morphism of sequences of filtered complexes from the tensor product of the sequences for $m$ and $n$ to the sequence for $m+n$: of the type of Lemma 3.3 (1) there for $\sigma^S$, and of Lemma 3.3 (2) for $\sigma^\Lambda$, exactly as in the proof of Proposition 3.4 there. Naturality (Lemma 2.1 (2)) and the filtered Lemma 3.3 give the two formulas. $\square$

3. Exercises

Exercise 3.1. Let $M$ be a bounded above complex of flat modules, and $n=1$. Show that the sequence of the proof of Theorem 2.2 is $0\to M\to C(M)\to M[1]\to0$, and that the isomorphism $\sigma^S_M\colon$ $M[1]=$ $S^1(M[1])\to$ $\Lambda^1(M)[1]=$ $M[1]$ is the identity, with the convention $\delta=-p\circ q^{-1}$.

Exercise 3.2. Let $M\in D^-F(A)$ with $\operatorname{gr}^0M\cong E$, $\operatorname{gr}^1M\cong E'$ and $\operatorname{gr}^pM\cong0$ otherwise. Describe $\operatorname{gr}^p\sigma^S_M$ and $\operatorname{gr}^p\sigma^\Lambda_M$ for $n=2$ and $p=0,1,2$.

Exercise 3.3. Let $M\in D^-F(A)$ have graded pieces $E_0$, $E_1$, $E_2$ in weights $0,1,2$. List the multi-indices $\nu$ with $|\nu|=3$ and $w(\nu)=3$, and give the signs with which the maps $\tau\circ\bigotimes_i\sigma^\Lambda_{E_i}$ occur in $\operatorname{gr}^3\sigma^\Lambda_M$ for $n=3$.

Exercise 3.4. Let $E\in D^-G(A)$ be concentrated in weight $1$. Show that $\sigma^S_E$ and $\sigma^\Lambda_E$ are the shift isomorphisms of $E_1$, placed in weight $n$, with no sign. What happens if $E$ is concentrated in weights $0$ and $1$ with $E_0=E_1=K$, for $n=2$?

4. Complete solutions

Solution 3.1. For $n=1$ the first Koszul sequence of $0\to E\to F\to G\to0$ in total degree $1$ is $0\to\Lambda^1E\to\Lambda^0E\otimes S^1F\to S^1G\to0$, that is $0\to E\to F\to G\to0$. Applied to $\Gamma\,i$ of the cone sequence and after $\operatorname{Tot}N$ it is $0\to M\to C(M)\to M[1]\to0$, because $\operatorname{Tot}N\Gamma\,iX=X$ for every complex $X$. Lemma 2.1 (2) gives $\delta=-p\circ q^{-1}\colon M[1]\to M[1]$. Here $C(M)^k=M^k\oplus M^{k+1}$ with $d(y,x)=(dy+x,-dx)$, $f(y)=(y,0)$, $g(y,x)=x$, and $C(f)^k=C(M)^k\oplus M^{k+1}$ with $d((y,x),e)=((dy+x+e,-dx),-de)$. The map $s\colon M[1]\to C(f)$, $s(x)=((0,x),-x)$, is a morphism of complexes: $d\,s(x)=((0,-dx),dx)=s(-dx)$, and $-dx$ is the differential of $x$ in $M[1]$. As $q\circ s=1$ and $q$ is a quasi-isomorphism, $q^{-1}=s$ in $D(A)$, and $\delta(x)=-p(s(x))=-(-x)=x$. So $\sigma^S_M$ is the identity.

Solution 3.2. The multi-indices with $|\nu|=2$ supported in $\{0,1\}$ are $2\delta_0$, $\delta_0+\delta_1$, $2\delta_1$, with weights $0,1,2$ and $e(\nu)=0,1,0$. By Proposition 2.3, $\operatorname{gr}^0\sigma^S_M=\sigma^S_E\colon LS^2(E[1])\to L\Lambda^2(E)[2]$, $\operatorname{gr}^2\sigma^S_M=\sigma^S_{E'}$, and $\operatorname{gr}^1\sigma^S_M=\tau\circ(\sigma^S_E\otimes\sigma^S_{E'})$ in degree $1$, which by Exercise 3.1 is $\tau\colon E[1]\otimes^{\mathbf L}E'[1]\to(E\otimes^{\mathbf L}E')[2]$, $x\otimes y\mapsto(-1)^{|x|}x\otimes y$. Likewise $\operatorname{gr}^0\sigma^\Lambda_M=\sigma^\Lambda_E$, $\operatorname{gr}^2\sigma^\Lambda_M=\sigma^\Lambda_{E'}$, and $\operatorname{gr}^1\sigma^\Lambda_M=-\tau$, the shift isomorphisms of degree $1$ being identities in both cases (Exercise 3.1 and its analogue for the second Koszul sequence, which in degree $1$ is the same sequence).

Solution 3.3. $\nu=(\nu_0,\nu_1,\nu_2)$ with $\nu_0+\nu_1+\nu_2=3$ and $\nu_1+2\nu_2=3$: either $\nu=3\delta_1$ or $\nu=\delta_0+\delta_1+\delta_2$. For $3\delta_1$, $e=0$, and the map is $\sigma^\Lambda_{E_1}\colon L\Lambda^3(E_1[1])\to L\Gamma^3(E_1)[3]$. For $\delta_0+\delta_1+\delta_2$, $e=1+1+1=3$, and the map is $-\tau\colon$ $E_0[1]\otimes^{\mathbf L}E_1[1]\otimes^{\mathbf L}E_2[1]\to$ $(E_0\otimes^{\mathbf L}E_1\otimes^{\mathbf L}E_2)[3]$, the shift isomorphisms in degree $1$ being identities. For $\sigma^S_M$ the same multi-indices occur, without signs.

Solution 3.4. The only multi-index of weight $n$ supported in $\{1\}$ is $n\delta_1$, with $e(n\delta_1)=0$; so $\sigma^S_E$ and $\sigma^\Lambda_E$ are $\sigma^S_{E_1}$ and $\sigma^\Lambda_{E_1}$ in weight $n$. For $E_0=E_1=K$ and $n=2$: in weights $0$ and $2$, $\sigma^S_E$ is $\sigma^S_K\colon LS^2(K[1])\to L\Lambda^2(K)[2]$, and in weight $1$ it is $\tau\colon K[1]\otimes^{\mathbf L}K[1]\to(K\otimes^{\mathbf L}K)[2]$; $\sigma^\Lambda_E$ is $\sigma^\Lambda_K$ in weights $0$ and $2$ and $-\tau$ in weight $1$. The underlying morphisms are the shift isomorphisms of $K\oplus K$, and the signs in weight $1$ are those of Proposition 3.4 of the lesson on Koszul complexes for $m=n=1$.

Sources

The filtered Koszul complexes and the shift isomorphisms on filtered and graded derived categories are due to Illusie, in his work on the cotangent complex, where they are used to construct Chern classes of perfect complexes; Proposition 2.3 states their compatibility with graded pieces, with the signs. Koszul complexes and the triangles of short exact sequences of complexes are as in the lesson on Koszul complexes and in the AI-integrated Stacks project, Chapter Derived Categories.