Diagonal maps of symmetric and exterior powers

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

For modules \(L\) and \(M\) over a commutative ring of a topos, multiplying elementary tensors factor by factor gives homomorphisms of algebras \[ \begin{gathered}\varphi\colon S(L\otimes M)\to\bigoplus_nS^nL\otimes S^nM,\\ \psi\colon\Lambda(L\otimes M)\to\bigoplus_n\Lambda^nL\otimes S^nM,\end{gathered} \] with \(\varphi((x_1\otimes y_1)\cdots(x_n\otimes y_n))=\) \(x_1\cdots x_n\otimes y_1\cdots y_n\) and the same formula for \(\psi\). Applied to an endomorphism twisted by a complex, \(E\to E\otimes^{\mathbf L}M\), the maps \(\varphi^n\) turn the symmetric powers of the endomorphism into endomorphisms of \(LS^nE\) twisted by \(LS^nM\), whose traces are the coefficients of the L-series and the Chern classes. This lesson constructs these maps and proves their properties:

Prerequisites. Filtered and graded powers of complexes (the conventions, Lemmas 1.2, 2.3, Propositions 1.5, 1.6, 2.2, 2.4, 2.5, 2.7, Theorem 2.6); Filtered Koszul complexes and the shift isomorphisms (Lemma 2.1, Theorem 2.2); Koszul complexes and the shift formulas (Construction 1.3, Theorem 2.1, Lemma 3.1, Theorem 3.2); Non-additive functors of complexes in a ringed topos (Sections 2 and 5); Filtered tensor products on a ringed topos (Lemmas 1.2, 1.4, 2.2, Proposition 3.2); Filtered derived categories (Lemma 3.1); The cotangent complex, Theorems 1.2 and 1.3 (Dold–Kan, Alexander–Whitney and shuffle maps). From the AI-integrated Stacks project: bounded above complexes of flat modules are K-flat [Stacks, Tag 06YQ].

Conventions

As in the lesson on filtered and graded powers of complexes: \(A\) is a commutative ring of a topos \(T\), \(\otimes=\otimes_A\), filtered modules have finite filtrations and their powers carry the filtrations of Definition 1.1 there, \(c\) is the comparison map (1.1) there and \(\mu\) the map (1.1) of the lesson on filtered tensor products. A complex concentrated in degrees \(\leq0\) is regarded as a chain complex in degrees \(\geq0\), \(X_k=X^{-k}\), and for such complexes \(\mathbf T(X)=N\mathbf T(\Gamma X)\), with \(\Gamma\) the inverse of normalization (Section 2.1 of the lesson on non-additive functors). For simplicial modules \(V\), \(W\), \(\operatorname{sh}\colon NV\otimes NW\to N(V\otimes W)\) and \(\operatorname{AW}\colon N(V\otimes W)\to NV\otimes NW\) are the shuffle and Alexander–Whitney maps, natural, associative and inverse to each other in the derived category (Theorem 1.3 of The cotangent complex); for complexes in degrees \(\leq0\) the map (5.1) of the lesson on non-additive functors is the shuffle map. \(D^{\leq0}F(A)\) and \(D^{\leq0}G(A)\) are the full subcategories of objects whose graded pieces have no cohomology in positive degrees.

1. Diagonal maps of modules

Definition 1.1. For modules \(L\), \(M\), put \(S_{\mathrm{diag}}(L,M)=\bigoplus_{n\geq0}S^nL\otimes S^nM\), a subalgebra of the commutative algebra \(SL\otimes SM\), and \(\Lambda S(L,M)=\bigoplus_{n\geq0}\Lambda^nL\otimes S^nM\), a subalgebra of \(\Lambda L\otimes SM\), whose product is \((a\otimes b)(a'\otimes b')=aa'\otimes bb'\).

Proposition 1.2. There are unique homomorphisms of graded algebras \[ \begin{gathered}\varphi\colon S(L\otimes M)\to S_{\mathrm{diag}}(L,M),\\ \psi\colon\Lambda(L\otimes M)\to\Lambda S(L,M)\end{gathered} \] that are the identity of \(L\otimes M\) in degree \(1\). They are natural in \(L\) and \(M\), and \[ \begin{aligned}&\varphi^n\bigl((x_1\otimes y_1)\cdots(x_n\otimes y_n)\bigr)\\ &\quad =x_1\cdots x_n\otimes y_1\cdots y_n,\\ &\psi^n\bigl((x_1\otimes y_1)\wedge\cdots\wedge(x_n\otimes y_n)\bigr)\\ &\quad =x_1\wedge\cdots\wedge x_n\otimes y_1\cdots y_n.\end{aligned} \]

Proof. We argue on sections and sheafify. \(S(L\otimes M)\) is the free commutative algebra on \(L\otimes M\), and \(S_{\mathrm{diag}}(L,M)\) is commutative. For \(\psi\), by the universal property of the exterior algebra it suffices that every \(z=\sum_kx_k\otimes y_k\) in \(L\otimes M\) has square zero in \(\Lambda S(L,M)\): \(z^2=\sum_{k,l}x_k\wedge x_l\otimes y_ky_l=0\), since the terms \((k,l)\) and \((l,k)\) cancel and \(x_k\wedge x_k=0\). The formulas follow from multiplicativity, and naturality from uniqueness. \(\square\)

Proposition 1.3. Let \(L\), \(M\) be filtered modules, and give \(L\otimes M\) the tensor filtration and \(S^nL\otimes S^nM\), \(\Lambda^nL\otimes S^nM\) the tensor products of the filtrations of the factors.

  1. \(\varphi^n\) and \(\psi^n\) are filtered. For graded \(L\), \(M\) they are graded.
  2. The maps \(c\circ S(\mu)\colon S(\operatorname{gr}L\otimes\operatorname{gr}M)\to\) \(\operatorname{gr}S(L\otimes M)\) and \(\mu\circ(c\otimes c)\colon S_{\mathrm{diag}}(\operatorname{gr}L,\operatorname{gr}M)\to\) \(\operatorname{gr}S_{\mathrm{diag}}(L,M)\) satisfy \[ \operatorname{gr}\varphi\circ c\circ S(\mu)=\mu\circ(c\otimes c)\circ\varphi, \] and the analogous identity holds for \(\psi\). These maps and their analogues for \(\psi\) are isomorphisms when \(\operatorname{gr}L\) and \(\operatorname{gr}M\) are flat.
  3. Let \(L=\bigoplus_iL_i\) and \(M=\bigoplus_jM_j\) be graded. Through the exponential isomorphism \(S(L\otimes M)\cong\bigotimes_{(i,j)}S(L_i\otimes M_j)\), taken in a fixed order of the pairs, \(\varphi_{L,M}\) is the product in \(S_{\mathrm{diag}}(L,M)\) of the maps \(\varphi_{L_i,M_j}\) followed by the inclusions \(S_{\mathrm{diag}}(L_i,M_j)\to S_{\mathrm{diag}}(L,M)\). The same holds for \(\psi\).

Proof. (1) \(F^k(L\otimes M)\) is generated by the \(x\otimes y\) with \(x\in F^iL\), \(y\in F^jM\), \(i+j\geq k\), so \(F^kS^n(L\otimes M)\) is generated by the products \(\prod_l(x_l\otimes y_l)\) with \(x_l\in F^{i_l}L\), \(y_l\in F^{j_l}M\) and \(\sum_l(i_l+j_l)\geq k\) (Definition 1.1 of the lesson on filtered and graded powers). Their images \(\prod_lx_l\otimes\prod_ly_l\) lie in \(F^{\sum i_l}S^nL\otimes F^{\sum j_l}S^nM\). The same argument applies to \(\psi^n\) and to gradings.

(2) All maps are homomorphisms of algebras: \(c\) by Proposition 1.5 (2) there, \(\mu\) because the products of \(SL\otimes SM\) and \(\Lambda L\otimes SM\) carry no signs, and \(\operatorname{gr}\varphi\), \(\operatorname{gr}\psi\) as graded pieces of filtered homomorphisms. The two composites agree on the generators \(\bar x\otimes\bar y\in\operatorname{gr}L\otimes\operatorname{gr}M\), which both send to the class of \(x\otimes y\). If \(\operatorname{gr}L\) and \(\operatorname{gr}M\) are flat, \(\mu\colon\operatorname{gr}L\otimes\operatorname{gr}M\to\operatorname{gr}(L\otimes M)\) is an isomorphism (Lemma 1.4 of the lesson on filtered tensor products), \(\operatorname{gr}(L\otimes M)\) is flat, and the maps \(c\) are isomorphisms (Proposition 1.6 of the lesson on filtered and graded powers); \(\mu\colon\operatorname{gr}S^nL\otimes\operatorname{gr}S^nM\to\operatorname{gr}(S^nL\otimes S^nM)\) is an isomorphism by Lemma 1.4 again.

(3) Both sides are homomorphisms of algebras \(\bigotimes_{(i,j)}S(L_i\otimes M_j)\to S_{\mathrm{diag}}(L,M)\), the target being commutative, and they agree on each \(L_i\otimes M_j\). For \(\psi\) the target \(\Lambda S(L,M)\) is anticommutative in the exterior degree, and the exponential isomorphism of exterior algebras carries the same signs on both sides. \(\square\)

Proposition 1.4. Let \(0\to L'\xrightarrow{u}L\xrightarrow{v}L''\to0\) be an exact sequence of modules and \(M\) a module. For \(p+q=n\), let \[ \begin{aligned}&\lambda_{p,q} \colon\Lambda^p(L'\otimes M)\otimes S^q(L\otimes M)\\&\quad \to\Lambda^pL'\otimes S^qL\otimes S^nM\end{aligned} \] be \(\psi^p\otimes\varphi^q\), followed by the exchange of the two middle factors and the product \(S^pM\otimes S^qM\to S^nM\). Then the \(\lambda_{p,q}\) form a morphism from the part of total degree \(n\) of \(\operatorname{Kos}_\bullet(u\otimes1_M)\) to the part of total degree \(n\) of \(\operatorname{Kos}_\bullet(u)\), tensored with \(S^nM\); it is \(\psi^n\) on \(\Lambda^n(L'\otimes M)\), and together with \(\varphi^n\colon S^n(L''\otimes M)\to S^nL''\otimes S^nM\) it is compatible with the augmentations of Construction 1.3 of the lesson on Koszul complexes. In particular, when \(L'\), \(L\), \(L''\) and \(M\) are flat, it is a morphism between the exact Koszul sequences of \(0\to L'\otimes M\to L\otimes M\to L''\otimes M\to0\) and of \(0\to L'\to L\to L''\to0\) tensored with \(S^nM\). All this is natural in the sequence and in \(M\), and filtered (resp. graded) for filtered (resp. graded) data.

Proof. Let \(\alpha=z_1\wedge\cdots\wedge z_p\) with \(z_k=x_k\otimes m_k\), \(x_k\in L'\), and \(\beta=\prod_l(y_l\otimes m'_l)\), \(y_l\in L\); such elements generate. Then \(\lambda(\alpha\otimes\beta)=\) \(x_1\wedge\cdots\wedge x_p\otimes\) \(y_1\cdots y_q\otimes\) \(m_1\cdots m_pm'_1\cdots m'_q\). The Koszul differential of \(\alpha\otimes\beta\) is \(\sum_k(-1)^{k-1}z_1\wedge\cdots\widehat{z_k}\cdots\wedge z_p\otimes\) \((u(x_k)\otimes m_k)\beta\), whose image under \(\lambda\) is \(\sum_k(-1)^{k-1}x_1\wedge\cdots\widehat{x_k}\cdots\wedge x_p\otimes\) \(u(x_k)y_1\cdots y_q\otimes\) \(m_1\cdots m_pm'_1\cdots m'_q\): the factor \(m_k\) moves from the exterior part to the symmetric part, and the product in \(S^nM\) is the same. This is the Koszul differential of \(\operatorname{Kos}_\bullet(u)\), tensored with \(S^nM\), applied to \(\lambda(\alpha\otimes\beta)\). For \(q=0\), \(\lambda_{n,0}=\psi^n\). The augmentations are \(S^n(v\otimes1)\) and \(S^n(v)\otimes1\), and \((S^n(v)\otimes1)\circ\varphi^n_{L,M}=\varphi^n_{L'',M}\circ S^n(v\otimes1)\) by naturality. Exactness of the two sequences in the flat case is Theorem 2.1 there, with \(S^nM\) flat. Naturality and the filtered and graded statements follow from those of \(\varphi\), \(\psi\) and of the products. \(\square\)

2. Diagonal maps of connective complexes

Construction 2.1. Let \(P\), \(Q\) be complexes of flat modules vanishing in degrees \(>0\). By Dold–Kan (Theorem 1.2 of the lesson on the cotangent complex), there is a unique morphism of simplicial modules \[ g_{P,Q}\colon\Gamma(P\otimes Q)\to\Gamma P\otimes\Gamma Q \] whose normalization is the shuffle map \(P\otimes Q=\) \(N\Gamma P\otimes N\Gamma Q\to\) \(N(\Gamma P\otimes\Gamma Q)\), namely \(\Gamma(\operatorname{sh})\) followed by the isomorphism \(\Gamma N\cong1\). Put \[ \begin{aligned}&\varphi^n_{P,Q} =\operatorname{AW}\circ N(\varphi^n)\circ NS^n(g_{P,Q})\\ &\quad \colon S^n(P\otimes Q)\\ &\quad \to S^n(P)\otimes S^n(Q),\\ &\psi^n_{P,Q} =\operatorname{AW}\circ N(\psi^n)\circ N\Lambda^n(g_{P,Q})\\ &\quad \colon\Lambda^n(P\otimes Q)\\ &\quad \to\Lambda^n(P)\otimes S^n(Q),\end{aligned} \] with \(\varphi^n\), \(\psi^n\) applied termwise to \(\Gamma P\otimes\Gamma Q\). These are morphisms of complexes, natural in \(P\) and \(Q\). For filtered (resp. graded) \(P\), \(Q\) they are filtered (resp. graded): the shuffle map is natural, so it maps \(NF^i\Gamma P\otimes NF^j\Gamma Q\) into \(N(F^i\Gamma P\otimes F^j\Gamma Q)\), hence \(g_{P,Q}\) is filtered; \(\varphi^n\), \(\psi^n\) are filtered termwise (Proposition 1.3); and \(\operatorname{AW}\) is natural.

Proposition 2.2. For \(L,M\in D^{\leq0}F(A)\), Construction 2.1 applied to filtered quasi-isomorphisms \(P\to L\), \(Q\to M\) from complexes with flat graded pieces vanishing in degrees \(>0\) defines morphisms in \(DF(A)\) \[ \begin{gathered}\varphi^n_{L,M} \colon LS^n(L\otimes^{\mathbf L}M)\\ \to LS^nL\otimes^{\mathbf L}LS^nM,\\ \psi^n_{L,M} \colon L\Lambda^n(L\otimes^{\mathbf L}M)\\ \to L\Lambda^nL\otimes^{\mathbf L}LS^nM,\end{gathered} \] independent of the choices and natural in \(L\) and \(M\). Their underlying morphisms are those of Construction 2.1 for the underlying complexes, and the same construction applies in \(D^{\leq0}G(A)\) and \(D^{\leq0}(A)\). Under the isomorphisms \(c\) of Theorem 2.6 (2) of the lesson on filtered and graded powers and (3.1) of the lesson on filtered tensor products, \(\operatorname{gr}\varphi^n_{L,M}=\varphi^n_{\operatorname{gr}L,\operatorname{gr}M}\) and \(\operatorname{gr}\psi^n_{L,M}=\psi^n_{\operatorname{gr}L,\operatorname{gr}M}\).

Proof. Such resolutions exist: \(L\) is isomorphic to \(\tau_{\leq0}\) of a representative (Lemma 3.1 of the lesson on filtered derived categories), and Proposition 2.4 of the lesson on filtered and graded powers applies with \(c=0\). The argument of Proposition 2.5 there, with the bound \(0\), identifies \(D^{\leq0}F(A)\) with the localization of the homotopy category of these complexes at the filtered quasi-isomorphisms. The complex \(P\otimes Q\) vanishes in degrees \(>0\), has flat graded pieces (Lemma 1.4 of the lesson on filtered tensor products) and represents \(L\otimes^{\mathbf L}M\) (Proposition 3.2 there), so \(S^n(P\otimes Q)\) represents \(LS^n(L\otimes^{\mathbf L}M)\) (Theorem 2.6 of the lesson on filtered and graded powers); and \(S^n(P)\otimes S^n(Q)\) represents \(LS^nL\otimes^{\mathbf L}LS^nM\), the factors being bounded above with flat graded pieces, hence filtered K-flat [Tag 06YQ]. Filtered quasi-isomorphisms \(P'\to P\), \(Q'\to Q\) induce filtered quasi-isomorphisms on both sides (Lemma 2.3 there and Lemma 2.2 (3) of the lesson on filtered tensor products), and the squares with \(\varphi^n\) commute by naturality; filtered homotopic maps induce equal morphisms in \(DF(A)\) by the cylinder argument of the proof of Theorem 2.6 there. This gives the morphisms and their independence. The underlying statement is clear.

For the graded pieces, the termwise maps \(c\) are natural and multiplicative, the shuffle and Alexander–Whitney maps are natural, and \(\operatorname{gr}\) commutes with \(\Gamma\) and \(N\); so \(\operatorname{gr}g_{P,Q}\) corresponds to \(g_{\operatorname{gr}P,\operatorname{gr}Q}\) through \(\mu\), and Proposition 1.3 (2), applied termwise, identifies \(\operatorname{gr}\varphi^n_{P,Q}\) with \(\varphi^n_{\operatorname{gr}P,\operatorname{gr}Q}\) through \(c\) and \(\mu\). The same holds for \(\psi\). \(\square\)

Proposition 2.3 (products). Let \(P\), \(Q\) be complexes of flat modules vanishing in degrees \(>0\).

  1. For simplicial modules \(V\), \(W\), \(v\in N_pV\), \(w\in N_qW\), and \(t\colon V\otimes W\to W\otimes V\) the termwise exchange of factors, \(\operatorname{sh}(w\otimes v)=\) \((-1)^{pq}N(t)(\operatorname{sh}(v\otimes w))\). Consequently the products of \(S(P)\) are graded commutative, and the shuffle map \(NV\otimes NW\to N(V\otimes W)\) is a homomorphism of differential graded algebras when \(V\), \(W\) are simplicial commutative algebras and \(NV\otimes NW\) carries the product \((a\otimes b)(a'\otimes b')=\) \((-1)^{|b||a'|}aa'\otimes bb'\).
  2. Give \(S_{\mathrm{diag}}(P,Q)=\bigoplus_nS^n(P)\otimes S^n(Q)\) the product \((a\otimes b)(a'\otimes b')=\) \((-1)^{|b||a'|}aa'\otimes bb'\). Then \(\varphi=\bigoplus_n\varphi^n_{P,Q}\) is multiplicative in the derived category: \(\varphi\circ(\text{product})=\) \((\text{product})\circ(\varphi\otimes\varphi)\) as morphisms \(S^m(P\otimes Q)\otimes S^n(P\otimes Q)\to\) \(S^{m+n}(P)\otimes S^{m+n}(Q)\). The same holds for \(\psi\) and \(\bigoplus_n\Lambda^n(P)\otimes S^n(Q)\), with the same sign rule.
  3. Let \(L,M\in D^{\leq0}G(A)\) have components \(L_i\), \(M_j\). Decompose \(LS^n(L\otimes^{\mathbf L}M)\) through \(L\otimes^{\mathbf L}M=\bigoplus_{(i,j)}L_i\otimes^{\mathbf L}M_j\), the pairs in lexicographic order, into the summands \(\bigotimes^{\mathbf L}_{(i,j)}LS^{\rho_{ij}}(L_i\otimes^{\mathbf L}M_j)\), \(\sum\rho_{ij}=n\) (Proposition 2.2 (2) and Theorem 2.6 of the lesson on filtered and graded powers, for this finite decomposition). Then \(\varphi^n_{L,M}\) restricted to such a summand is the composite of \(\bigotimes_{(i,j)}\varphi^{\rho_{ij}}_{L_i,M_j}\), of the symmetry \(\bigotimes_{(i,j)}(LS^{\rho_{ij}}L_i\otimes^{\mathbf L}LS^{\rho_{ij}}M_j)\cong\) \(\bigl(\bigotimes_{(i,j)}LS^{\rho_{ij}}L_i\bigr)\otimes\) \(^{\mathbf L}\bigl(\bigotimes_{(i,j)}LS^{\rho_{ij}}M_j\bigr)\) of the symmetric monoidal category \(D(A)\), and of the tensor product of the iterated products \(\bigotimes_{(i,j)}LS^{\rho_{ij}}L_i\to LS^nL\) and \(\bigotimes_{(i,j)}LS^{\rho_{ij}}M_j\to LS^nM\) of the maps induced by the inclusions \(L_i\to L\), \(M_j\to M\). The same holds for \(\psi^n\), with \(L\Lambda\) in place of the first \(LS\). In particular, for \(M\) concentrated in weight \(0\), \(\varphi^n_{L,M}\) maps the summand \(\bigotimes^{\mathbf L}_iLS^{\nu_i}(L_i\otimes^{\mathbf L}M)\) into \(\bigl(\bigotimes^{\mathbf L}_iLS^{\nu_i}L_i\bigr)\otimes^{\mathbf L}LS^nM\) by \(\bigotimes_i\varphi^{\nu_i}_{L_i,M}\), the symmetry, and the product \(\bigotimes_iLS^{\nu_i}M\to LS^nM\).

Proof. (1) In the formula of Theorem 1.3 of the lesson on the cotangent complex, \(\operatorname{sh}(v\otimes w)=\sum_{(\mu,\nu)}\pm s_\nu v\otimes s_\mu w\) over the \((p,q)\)-shuffles \((\mu,\nu)\), with the sign of the shuffle. Exchanging the roles of \(v\) and \(w\) exchanges the blocks \(\mu\) and \(\nu\); the shuffle \((\nu,\mu)\) is the shuffle \((\mu,\nu)\) composed with the exchange of a block of \(p\) and a block of \(q\) elements, of sign \((-1)^{pq}\). For \(S(P)\), the product of \(x\in S^m(P)_p\) and \(y\in S^n(P)_q\) is \(N(\text{mult})(\operatorname{sh}(x\otimes y))\), and the termwise multiplication satisfies \(\text{mult}\circ t=\text{mult}\); so \(yx=(-1)^{pq}xy\). For the last assertion, the product of \(N(V\otimes W)\) is \(N(\text{mult}_{V\otimes W})\circ\operatorname{sh}\), and \(\text{mult}_{V\otimes W}=\) \((\text{mult}_V\otimes\text{mult}_W)\circ(1\otimes t\otimes1)\) termwise. By the associativity and naturality of the shuffle map and the first assertion applied to the middle exchange, \(\operatorname{sh}\bigl((a\otimes b)(a'\otimes b')\bigr)=\) \(\operatorname{sh}(a\otimes b)\cdot\operatorname{sh}(a'\otimes b')\).

(2) The termwise map \(\Phi=\) \(\varphi\circ S(g_{P,Q})\colon\) \(S(\Gamma(P\otimes Q))\to\) \(S_{\mathrm{diag}}(\Gamma P,\Gamma Q)\) is a homomorphism of simplicial algebras, so \(N(\Phi)\) is multiplicative for the products defined by the shuffle map, by its naturality. In the derived category \(\operatorname{AW}\) is the inverse of the shuffle map \(NS(\Gamma P)\otimes NS(\Gamma Q)\to\) \(N(S(\Gamma P)\otimes S(\Gamma Q))\), which is multiplicative by (1), and \(S_{\mathrm{diag}}(\Gamma P,\Gamma Q)\) is a simplicial subalgebra of \(S(\Gamma P)\otimes S(\Gamma Q)\). Composing gives the statement for \(\varphi\); the argument for \(\psi\) is the same, the products of \(\Lambda(\Gamma P)\otimes S(\Gamma Q)\) carrying no sign termwise.

(3) Represent \(L\), \(M\) by graded complexes \(P=\bigoplus_iP_i\), \(Q=\bigoplus_jQ_j\) as in Proposition 2.2. As in the proof of Proposition 2.3 (2) of the lesson on filtered Koszul complexes, the summand of \(\rho\) is embedded in \(S^n(P\otimes Q)\) by the iterated product of the maps \(S^{\rho_{ij}}(\iota_i\otimes\iota'_j)\), \(\iota_i\colon P_i\to P\), \(\iota'_j\colon Q_j\to Q\). By (2) and the naturality of \(\varphi\) in both variables, \(\varphi^n\) of this product is the iterated product in \(S_{\mathrm{diag}}(P,Q)\) of the elements \((S^{\rho_{ij}}(\iota_i)\otimes S^{\rho_{ij}}(\iota'_j))\varphi^{\rho_{ij}}_{P_i,Q_j}(\cdot)\). An iterated product \((a_1\otimes b_1)\cdots(a_r\otimes b_r)\) in \(S_{\mathrm{diag}}\) is \(\pm(a_1\cdots a_r)\otimes(b_1\cdots b_r)\), with the sign \(\prod_{k<l}(-1)^{|b_k||a_l|}\) of the symmetry of Lemma 1.2 of the lesson on filtered tensor products. The same argument applies to \(\psi\). When \(M\) is concentrated in weight \(0\), the products \(a_1\cdots a_r\) of the images of the \(S^{\nu_i}(P_i)\) form the summand of \(\nu\) of \(S^n(P)\). \(\square\)

Proposition 2.4 (diagonal maps and shift isomorphisms). Let \(L,M\in D^{\leq0}F(A)\) (or \(D^{\leq0}G(A)\), or \(D^{\leq0}(A)\)) and \(n\geq1\). Then \[ (\sigma^S_L\otimes1)\circ\varphi^n_{L[1],M}=\psi^n_{L,M}[n]\circ\sigma^S_{L\otimes^{\mathbf L}M} \] as morphisms \(LS^n((L\otimes^{\mathbf L}M)[1])\to\) \((L\Lambda^nL\otimes^{\mathbf L}LS^nM)[n]\). Here \(\sigma^S\) are the shift isomorphisms of Theorem 2.2 of the lesson on filtered Koszul complexes, and \(L[1]\otimes^{\mathbf L}M=(L\otimes^{\mathbf L}M)[1]\), \(L\Lambda^n(L)[n]\otimes^{\mathbf L}LS^nM=\) \((L\Lambda^nL\otimes^{\mathbf L}LS^nM)[n]\) without sign, the shift being on the first factor.

Proof. Represent \(L\), \(M\) by \(P\), \(Q\) as in Proposition 2.2. Let \(E=(0\to P\to C(P)\to P[1]\to0)\) be the cone sequence. Tensored with \(Q\) on the right it is the cone sequence of \(P\otimes Q\): \((y,x)\otimes q\mapsto(y\otimes q,x\otimes q)\) is an isomorphism \(C(P)\otimes Q\to C(P\otimes Q)\), and \(P[1]\otimes Q=(P\otimes Q)[1]\), the signs of the differentials agreeing because the shifted factor is on the left. The maps \(g_{X,Q}\) are natural in \(X\), so they form a morphism of degreewise split exact sequences of simplicial flat modules from \(\Gamma(E\otimes Q)\) to \(\Gamma E\otimes\Gamma Q\) (termwise tensor product). The first Koszul sequences in total degree \(n\) are natural in the exact sequence, which gives a morphism of sequences from the Koszul sequence of \(\Gamma(E\otimes Q)\) to that of \(\Gamma E\otimes\Gamma Q\). Proposition 1.4, applied termwise with \(M=(\Gamma Q)_m\), gives a morphism from the latter to the Koszul sequence of \(\Gamma E\), tensored termwise with \(S^n(\Gamma Q)\). Apply \(N\), and then \(\operatorname{AW}\), which is natural: the result is a morphism of sequences \[ \begin{aligned}&N\bigl(\text{Koszul sequence of }\Gamma(E\otimes Q)\bigr)\\&\quad \to N\bigl(\text{Koszul sequence of }\Gamma E\bigr)\\&\quad \otimes S^n(Q).\end{aligned} \] The source is the sequence that defines \(\sigma^S_{P\otimes Q}\) (proof of Theorem 2.2 there). The target is the sequence that defines \(\sigma^S_P\), tensored with \(S^n(Q)\); its intermediate terms are acyclic, the complex \(S^n(Q)\) being bounded above with flat terms [Tag 06YQ], and its isomorphism is \(\sigma^S_P\otimes1\), because the triangle of a short exact sequence tensored on the right with \(S^n(Q)\) is the tensor product of the triangle with \(S^n(Q)\) (the cone and the maps \(r\), \(p\) tensor with the identity, without sign). On the last terms the morphism is \(\operatorname{AW}\circ N(\varphi^n)\circ NS^n(g_{P[1],Q})=\varphi^n_{P[1],Q}\), and on the first terms \(\operatorname{AW}\circ N(\psi^n)\circ N\Lambda^n(g_{P,Q})=\psi^n_{P,Q}\). The naturality of Lemma 2.1 (2) of the lesson on filtered Koszul complexes (Lemma 3.1 (1) of the lesson on Koszul complexes in the unfiltered case) gives the equality. In the filtered and graded cases every map is filtered or graded. \(\square\)

Remark 2.5 (complexes with terms in positive degrees). For objects of \(D^-F(A)\) that are not concentrated in degrees \(\leq0\), the functors \(L\mathbf T^n\) are computed through the mixed simplicial–cosimplicial objects \(\Gamma\,iP\) of Section 2.2 of the lesson on non-additive functors. Applied termwise to \(\Gamma\,iP\otimes\Gamma\,iQ\), the maps \(\varphi^n\) give morphisms \(\operatorname{Tot}NS^n(\Gamma\,iP\otimes\Gamma\,iQ)\to\) \(S^n(P)\otimes S^n(Q)\) in the derived category. The mixed object \(\Gamma\,iP\otimes\Gamma\,iQ\) is not \(\Gamma\,i(P\otimes Q)\): when \(P^a\neq0\) for some \(a<0\) and \(Q^b\neq0\) for some \(b>0\), the double complex \(iP\otimes iQ\) has nonzero terms off the two axes. Whether \(\operatorname{Tot}NS^n(\Gamma\,iP\otimes\Gamma\,iQ)\) represents \(LS^n(P\otimes Q)\) in that case is not proved in this course, and the morphisms of this section are defined for objects concentrated in degrees \(\leq0\) only.

3. Exercises

Exercise 3.1. Let \(L=Ae_1\oplus Ae_2\) and \(M=Af_1\oplus Af_2\). Show that \(\varphi^2\colon S^2(L\otimes M)\to S^2L\otimes S^2M\) is surjective and that its kernel is free of rank one, generated by \((e_1\otimes f_1)(e_2\otimes f_2)-(e_1\otimes f_2)(e_2\otimes f_1)\).

Exercise 3.2. With \(L\), \(M\) as in Exercise 3.1, compute \(\psi^2\colon\Lambda^2(L\otimes M)\to\Lambda^2L\otimes S^2M\) on the basis \((e_i\otimes f_k)\wedge(e_j\otimes f_l)\), and show that it is surjective with kernel free of rank three.

Exercise 3.3. Let \(L\in D^{\leq0}(A)\) and \(M=A\). Show that \(\varphi^n_{L,A}\) is the canonical isomorphism \(LS^n(L)\cong LS^n(L)\otimes^{\mathbf L}A\) and that \(\psi^n_{L,A}\) is the canonical isomorphism \(L\Lambda^n(L)\cong L\Lambda^n(L)\otimes^{\mathbf L}A\), and that Proposition 2.4 is then a tautology.

4. Complete solutions

Solution 3.1. Write \(z_{ik}=e_i\otimes f_k\). \(S^2(L\otimes M)\) is free on the ten monomials \(z_{ik}z_{jl}\), and \(S^2L\otimes S^2M\) on the nine products \(e_ie_j\otimes f_kf_l\) with \(i\leq j\), \(k\leq l\). Since \(\varphi^2(z_{ik}z_{jl})=e_ie_j\otimes f_kf_l\), each basis element \(e_ie_j\otimes f_kf_l\) is the image of \(z_{ik}z_{jl}\), so \(\varphi^2\) is surjective. Two monomials have the same image exactly when they are \(z_{11}z_{22}\) and \(z_{12}z_{21}\), both mapped to \(e_1e_2\otimes f_1f_2\); all other images are distinct basis elements. So the kernel consists of the multiples of \(z_{11}z_{22}-z_{12}z_{21}\), which is a basis element of a direct summand of rank one.

Solution 3.2. \(\Lambda^2(L\otimes M)\) is free on the six products \(z_{ik}\wedge z_{jl}\) with \((i,k)<(j,l)\) lexicographically, and \(\Lambda^2L\otimes S^2M\) is free on \(e_1\wedge e_2\otimes f_1^2\), \(e_1\wedge e_2\otimes f_1f_2\), \(e_1\wedge e_2\otimes f_2^2\). By Proposition 1.2, \(\psi^2(z_{ik}\wedge z_{jl})=e_i\wedge e_j\otimes f_kf_l\), which is \(0\) when \(i=j\). So \(z_{11}\wedge z_{12}\) and \(z_{21}\wedge z_{22}\) map to \(0\); \(z_{11}\wedge z_{21}\mapsto e_1\wedge e_2\otimes f_1^2\); \(z_{12}\wedge z_{22}\mapsto e_1\wedge e_2\otimes f_2^2\); and \(z_{11}\wedge z_{22}\), \(z_{12}\wedge z_{21}\) both map to \(e_1\wedge e_2\otimes f_1f_2\). Hence \(\psi^2\) is surjective, and its kernel is free on \(z_{11}\wedge z_{12}\), \(z_{21}\wedge z_{22}\) and \(z_{11}\wedge z_{22}-z_{12}\wedge z_{21}\).

Solution 3.3. For \(M=A\), \(S_{\mathrm{diag}}(L,A)=\bigoplus_nS^nL\otimes S^nA\) and \(S^nA=A\) with basis \(1^n\); \(\varphi^n(\prod(x_k\otimes1))=\prod x_k\otimes1\), which is the canonical isomorphism \(S^n(L\otimes A)\cong S^nL\otimes A\); likewise for \(\psi^n\). On complexes, \(\Gamma A\) is the constant simplicial module \(A\), \(g_{P,A}\) is the canonical isomorphism \(\Gamma(P\otimes A)\cong\Gamma P\otimes A\) (the shuffle map is the identity when one factor is in degree \(0\)), and \(\operatorname{AW}\) is the identity in that case. So \(\varphi^n_{L,A}\) and \(\psi^n_{L,A}\) are the canonical isomorphisms, and Proposition 2.4 reduces to \(\sigma^S_L=\sigma^S_L\).

Sources

The diagonal maps \(\varphi\) and \(\psi\), their compatibility with filtrations, gradings and shift isomorphisms, and their use for the L-series and Chern classes of perfect complexes are due to Illusie, in his work on the cotangent complex, where they are stated on the whole bounded above filtered derived category. The shuffle and Alexander–Whitney maps are due to Eilenberg and Mac Lane.