Spectral sequences for derived tensor products

Written by Claude Opus 5.5 (Anthropic) on 20 October 2026. Self-checked by the writing AI. Public domain (CC0).

The homology of a derived tensor product $E\otimes^\ell_AF$ of modules over a simplicial ring $A$ is computed by three spectral sequences. The first starts from the Tor groups of the modules $E_p$ and $F_p$ over the rings $A_p$, degree by degree. The second, a Künneth spectral sequence, starts from Tor over the graded ring $H_*(A)$ of the homology modules $H_*(E)$ and $H_*(F)$. The third starts from the derived tensor products of the homology modules of $E$, regarded as modules over $\pi_0(A)$; it comes from a filtration of $E$ by submodules whose normalized complexes are the truncations of that of $E$. This lesson constructs the three, together with the products on homology given by shuffle maps on which the Künneth spectral sequence rests. The Künneth spectral sequence is one of modules over the homology ring of $A$, from its first page to its abutment. The second and third depend only on the quasi-isomorphism classes of $E$ and $F$ from their second page on. The first does not, as an example shows.

Prerequisites. Derived tensor products of simplicial modules; Triangles and Ext groups of simplicial modules; Simplicial rings and modules in a topos, Proposition 2.1; Lemma 2.6 of Multisimplicial objects and spectral diagrams; Theorems 1.1–1.3 of The cotangent complex. From the AI-integrated Stacks project: the spectral sequences of a double complex and of a filtered complex [Stacks, Tag 0132, Tag 012M, Tag 012W]; flat modules [Tag 03EX, Tag 03EY]; the Dold–Kan correspondence [Tag 019G].

Conventions

Notation is that of Derived tensor products of simplicial modules: $T$ is a topos, $A$ a simplicial ring, $P(L)\to L$ the flat replacement of an $A$-module and $F_A(L)$ the standard free resolution, a bisimplicial module whose external columns resolve the terms of $L$. For an $A$-module $E$, $H_*(E)=\bigoplus_n\pi_n(E)$ is a graded module over the graded commutative ring $H_*(A)$ (Proposition 2.1 of the lesson on simplicial rings and modules); in particular each $\pi_n(E)$ is a $\pi_0(A)$-module. The map $A_n\to A_0\to\pi_0(A)$, given by the vertex $0$ and the projection, is a map from $A$ to the constant simplicial ring $\pi_0(A)$, through which every $\pi_0(A)$-module is an $A$-module, constant as a simplicial object. The ring $A_0$ acts on every $A$-module through the degeneracies $A_0\to A_n$, compatibly with all simplicial operators; on homology this action is that of $\pi_0(A)\subseteq H_*(A)$, because the shuffle product with an element of degree $0$ is multiplication by its degeneracy. Spectral sequences are homological: $d^r\colon E^r_{p,q}\to E^r_{p-r,q+r-1}$. The results on filtered and double complexes are quoted from the Stacks project in cohomological form and used after the change of indices $n\mapsto-n$.

1. Shuffle maps on homology

Proposition 1.1. Let $L$ and $M$ be $A$-modules. The shuffle map, followed by the projection $L\otimes_{\mathbb Z}M\to L\otimes_AM$, induces a chain map $NL\otimes_{NA}NM\longrightarrow N(L\otimes_AM),$ natural in $L$ and $M$, and hence a homomorphism of graded $H_*(A)$-modules $\begin{aligned}&H_*(L)\otimes_{H_*(A)}H_*(M)\\&\quad \longrightarrow H_*(L\otimes_AM).\end{aligned}$ Both are associative and unital in any number of factors.

Proof. The shuffle map $\operatorname{sh}\colon NL\otimes_{\mathbb Z}NM\to N(L\otimes_{\mathbb Z}M)$ is a natural chain map (Theorem 1.3 of The cotangent complex), and so is its composite $\phi$ with the projection to $N(L\otimes_AM)$. The module structures of $NL$ and $NM$ over the differential graded ring $NA$ are shuffle maps followed by the actions (Proposition 2.1 of the lesson on simplicial rings and modules). For $x\in NL$, $a\in NA$ and $y\in NM$, both $\phi(xa\otimes y)$ and $\phi(x\otimes ay)$ are images of the iterated shuffle $\operatorname{sh}(x\otimes a\otimes y)\in N(L\otimes A\otimes M)$, by associativity of shuffles, under the two maps $L\otimes_{\mathbb Z}A\otimes_{\mathbb Z}M\to L\otimes_AM$ given by the actions on $L$ and on $M$. These two maps agree. So $\phi$ factors through $NL\otimes_{NA}NM$. On homology it gives the second map. Associativity and units follow from those of the iterated shuffle. $\square$

2. The Tor spectral sequence

For $A$-modules $E$ and $F$ and $q\geq0$, let $\operatorname{Tor}^A_q(E,F)$ be the $A$-module whose term of degree $p$ is $\operatorname{Tor}^{A_p}_q(E_p,F_p)$. It is the external homology of the bisimplicial module $cE\otimes_AF_A(F)$: its column $p$ is $E_p\otimes_{A_p}F_A(F)_{p,\bullet}$, and $F_A(F)_{p,\bullet}$ is a resolution of $F_p$ by free $A_p$-modules (Lemma 1.3 of the lesson on derived tensor products). In particular it is functorial in $E$ and $F$.

Theorem 2.1. There is a spectral sequence, functorial in $E$ and $F$, $\begin{aligned}&E^2_{p,q} =H_p\bigl(\operatorname{Tor}^A_q(E,F)\bigr)\\&\quad \Longrightarrow H_{p+q}(E\otimes^\ell_AF).\end{aligned}$ From the second page on it is a spectral sequence of $\pi_0(A)$-modules.

Proof. The diagonal of $cE\otimes_AF_A(F)$ is $E\otimes_AP(F)$, which is $E\otimes^\ell_AF$ by Proposition 3.2 (1) of the lesson on derived tensor products, since $P(F)$ is flat. By Lemma 2.6 of the lesson on multisimplicial objects, its normalized complex is naturally homotopy equivalent to the total complex of the double normalization $K$ of $cE\otimes_AF_A(F)$, with internal degree $p$ and external degree $q$. Consider the spectral sequence of $K$ that first takes homology in the external direction [Stacks, Tag 0132]. Since normalization in the internal direction is a natural direct summand (Theorem 1.1 of The cotangent complex), it commutes with external homology. So its first page is the internal normalization of the external homology, $N_p\operatorname{Tor}^A_q(E,F)$, with the differential of $N\operatorname{Tor}^A_q(E,F)$, and its second page is $H_p(\operatorname{Tor}^A_q(E,F))$. $K$ lies in the first quadrant, so the spectral sequence converges to $H_*(\operatorname{Tot}K)=H_*(E\otimes^\ell_AF)$. Every construction is functorial and linear for the action of $A_0$ through the degeneracies, including the Eilenberg–Zilber maps, which are combinations of simplicial operators. On the second page and on the abutment this action is that of $\pi_0(A)$, and the later pages are subquotients of the second. $\square$

Remark 2.2 (invariance). The second page of Theorem 2.1 is not invariant under quasi-isomorphisms of $E$. Let $T$ be the topos of sets, $A=\mathbb Z$, $F=\mathbb Z/2$, and let $E\to\mathbb Z/2$ be the quasi-isomorphism from the free simplicial abelian group $E$ that corresponds under Dold–Kan to the complex $\mathbb Z\xrightarrow{2}\mathbb Z$ [Stacks, Tag 019G]. For $E$, every $E_p$ is free, so $\operatorname{Tor}_q=0$ for $q>0$, and $\operatorname{Tor}_0=E\otimes\mathbb Z/2$ corresponds to $\mathbb Z/2\xrightarrow{0}\mathbb Z/2$: the second page is $\mathbb Z/2$ at $(0,0)$ and $(1,0)$. For the constant module $\mathbb Z/2$, $\operatorname{Tor}_0$ and $\operatorname{Tor}_1$ are the constant module $\mathbb Z/2$: the second page is $\mathbb Z/2$ at $(0,0)$ and $(0,1)$. The two spectral sequences have the same abutment, $\operatorname{Tor}^{\mathbb Z}_*(\mathbb Z/2,\mathbb Z/2)$, but the map induced on the second page is not an isomorphism at $(1,0)$.

When $A$ is a ring $\Lambda$ placed in every degree, the abutment is the homology of $NE\otimes^{\mathbf L}_\Lambda NF$ (Proposition 3.4 of the lesson on derived tensor products), and the second page consists of the homology of the simplicial modules $\operatorname{Tor}^\Lambda_q(E_p,F_p)$.

3. The Postnikov spectral sequence

Definition 3.1. For an $A$-module $E$ and $q\geq0$, let $E^{[q]}\subseteq E$ consist of the simplices all of whose faces of dimension less than $q$ vanish: $E^{[q]}_n$ is the set of $x\in E_n$ with $\varphi^*x=0$ for every injective $\varphi\colon[m]\to[n]$ with $m<q$. In particular $E^{[q]}_n=0$ for $n<q$, and $E^{[0]}=E$.

Lemma 3.2. $E^{[q]}$ is an $A$-submodule of $E$, natural in $E$, and its normalized complex is the truncation $\begin{aligned}&N(E^{[q]}) =\tau_{\geq q}NE \colon\quad\cdots\\&\quad \to N_{q+2}E \to N_{q+1}E\\&\quad \to Z_q(NE) \to0.\end{aligned}$ The quotient $E^{[q]}/E^{[q+1]}$ has homotopy $\pi_q(E)$ in degree $q$ and no other homotopy.

Proof. For a simplicial operator $\theta\colon[n']\to[n]$ and an injective $\varphi\colon[m]\to[n']$ with $m<q$, factor $\theta\varphi$ as a surjection followed by an injection $[m']\to[n]$, with $m'\leq m<q$. Then $\varphi^*\theta^*x=0$ for $x\in E^{[q]}_n$. So $E^{[q]}$ is a simplicial subobject. It is stable under $A$, because $\varphi^*(ax)=\varphi^*(a)\varphi^*(x)$. Its normalized complex is $N_n(E^{[q]})=N_nE\cap E^{[q]}_n$. Every composite of faces can be written $d_{i_1}\cdots d_{i_k}$ with $i_1<\cdots<i_k$. For $x\in N_nE$, $d_ix=0$ when $i\geq1$, so the only proper face of $x$ that can be nonzero is $d_0x$. The element $d_0x$ lies in $N_{n-1}E$, and $d_0d_0x=d_0d_1x=0$, so $d_0x$ has no nonzero proper face. Hence $x\in E^{[q]}_n$ if and only if $n>q$, or $n=q$ and $d_0x=0$. This is the stated truncation. The normalized complex of the quotient is $N_{q+1}E/Z_{q+1}\to Z_q$ in degrees $q+1$ and $q$, whose only homology is $H_q(NE)=\pi_q(E)$. $\square$

Lemma 3.3. Let $G$ be an $A$-module with $\pi_n(G)=0$ for $n\neq q$. Then $G\cong\sigma^q\pi_q(G)$ in $D_\bullet(A)$, where $\pi_q(G)$ is the constant $A$-module through $A\to\pi_0(A)$.

Proof. First let $q=0$. Send $x\in G_n$ to the class in $\pi_0(G)$ of its vertex $0$. The vertices of a simplex are joined by its edges, so all have the same class, and the map commutes with faces and degeneracies. It is $A$-linear for the action of $A$ on $\pi_0(G)$ through $\pi_0(A)$, because the vertex of $ax$ is the product of the vertices. It induces the identity on $\pi_0$, so it is a quasi-isomorphism $G\to\pi_0(G)$. In general, $\Omega^qG$ has $\pi_n(\Omega^qG)=\pi_{n+q}(G)$, which vanishes for $n\neq0$, so $\Omega^qG\cong\pi_q(G)$, compatibly with the action of $\pi_0(A)$. Since $\pi_n(G)=0$ for $n<q$, the counit $\sigma^q\Omega^qG\to G$ is an isomorphism in $D_\bullet(A)$ (Proposition 2.2 of the lesson on triangles). So $G\cong\sigma^q\pi_q(G)$. $\square$

Theorem 3.4. There is a spectral sequence, functorial in $E$ and $F$, $\begin{aligned}&E^2_{p,q} =H_p\bigl(\pi_q(E)\otimes^\ell_AF\bigr)\\&\quad \Longrightarrow H_{p+q}(E\otimes^\ell_AF),\end{aligned}$ where $\pi_q(E)$ is the constant $A$-module through $A\to\pi_0(A)$. From the second page on it is a spectral sequence of $\pi_0(A)$-modules, and a quasi-isomorphism in either variable induces an isomorphism from the second page on.

Proof. Put $L=P(F)$, a flat $A$-module, so that $E\otimes_AL$ is $E\otimes^\ell_AF$ (Proposition 3.2 (1) of the lesson on derived tensor products). Tensoring with $L$ is exact [Stacks, Tag 03EX], so the submodules $E^{[q]}\otimes_AL$ form a decreasing filtration of $E\otimes_AL$ with quotients $(E^{[q]}/E^{[q+1]})\otimes_AL$. It induces a filtration of the normalized complex $N(E\otimes_AL)$ by subcomplexes, whose graded pieces are the normalized complexes of these quotients, since normalization is exact. In degree $n$ the filtration is finite: $(E^{[q]}\otimes L)_n=0$ for $q>n$. By [Stacks, Tags 012M, 012W], there is a convergent spectral sequence with first page $\begin{aligned}&H_n\bigl((E^{[q]}/E^{[q+1]})\otimes_AL\bigr)\\&\quad \Longrightarrow H_n(E\otimes_AL),\end{aligned}$ in filtration degree $q$ and total degree $n$. By Lemmas 3.2 and 3.3, $E^{[q]}/E^{[q+1]}\cong\sigma^q\pi_q(E)$ in $D_\bullet(A)$. The functor $-\otimes_AL$ computes $-\otimes^\ell_AF$ and commutes with $\sigma$ (Proposition 3.3 of the lesson on derived tensor products). So the first page in filtration degree $q$ and total degree $n$ is $H_{n-q}(\pi_q(E)\otimes^\ell_AF)$. Put $p=n-q$. The first differential goes from filtration degree $q$ to $q+1$ and lowers $n$ by one, so it maps $(p,q)$ to $(p-2,q+1)$. Shifting the numbering of the pages by one gives the stated spectral sequence, with $d^r\colon E^r_{p,q}\to E^r_{p-r,q+r-1}$. A quasi-isomorphism $E\to E'$ induces isomorphisms $\pi_q(E)\to\pi_q(E')$, hence an isomorphism on the second page, and a quasi-isomorphism $F\to F'$ induces an isomorphism on $\pi_q(E)\otimes^\ell_A-$. A map of spectral sequences that is an isomorphism on one page is one on all later pages. The filtration consists of $A$-submodules, so the spectral sequence is linear for $A_0$, which acts through $\pi_0(A)$ from the second page on. $\square$

4. Special modules and the Künneth spectral sequence

Graded modules over $H_*(A)$ are taken in the topos, with the signs of graded commutativity. A graded module is flat if tensoring with it is exact on graded modules. For graded modules $V$ and $W$, $\operatorname{Tor}^{H_*(A)}_p(V,W)$ is the $p$-th homology of $V\otimes_{H_*(A)}\Phi_\bullet$ for a resolution $\Phi_\bullet\to W$ by flat graded modules; it is graded by the internal degree, and we write $\operatorname{Tor}^{H_*(A)}_p(V,W)_q$ for its part of degree $q$. It does not depend on the resolution: if $\Psi_\bullet\to V$ is a flat resolution, the double complex $\Psi_\bullet\otimes\Phi_\bullet$ maps to $V\otimes\Phi_\bullet$ and to $\Psi_\bullet\otimes W$ by maps that are quasi-isomorphisms on every column, respectively row, because the terms of $\Phi$ and of $\Psi$ are flat. Both total complexes are then quasi-isomorphic, as in the proof of Lemma 1.1 of the lesson on derived tensor products. Flat resolutions exist, by sums of free modules on sheaves of sets in each degree.

Definition 4.1. An $A$-module $M$ is special if $M\cong A\otimes_{\mathbb Z}S$ for a simplicial abelian sheaf $S$ whose terms $S_n$ and whose homology sheaves $H_n(S)$ are flat abelian sheaves.

Lemma 4.2. Let $S$ be a complex of flat abelian sheaves in degrees $\geq0$ whose homology sheaves are flat. For every complex $X$ of abelian sheaves, the natural map $H_*(X)\otimes H_*(S)\to H_*(\operatorname{Tot}(X\otimes S))$ is an isomorphism.

Proof. Let $Z_n$ and $B_n$ be the cycles and boundaries of $S$. The sheaf $Z_0=S_0$ is flat. If $Z_n$ is flat, so is $B_n$, by the exact sequence $0\to B_n\to Z_n\to H_n(S)\to0$, and then so is $Z_{n+1}$, by $0\to Z_{n+1}\to S_{n+1}\to B_n\to0$ [Stacks, Tag 03EY]. So all $Z_n$ and $B_n$ are flat. The sequence of complexes $0\to Z\to S\to B[-1]\to0$, with zero differentials on $Z$ and on $B$, is exact, and stays exact after tensoring with $X$ because $B$ is flat [Stacks, Tag 03EX]. Since $Z_n$ and $B_n$ are flat, $H_*(X\otimes Z)=H_*(X)\otimes Z$ and $H_*(X\otimes B)=H_*(X)\otimes B$. The connecting maps of the long exact sequence are $1\otimes(B_n\subseteq Z_n)$, which is injective with cokernel $H_*(X)\otimes H_n(S)$, because $H_n(S)$ is flat [Stacks, Tag 03EX]. So $H_*(X\otimes S)\cong H_*(X)\otimes H_*(S)$, and the isomorphism is induced by the natural map. $\square$

Lemma 4.3. Let $M=A\otimes_{\mathbb Z}S$ be special. For every $A$-module $L$, the map $H_*(L)\otimes_{H_*(A)}H_*(M)\to H_*(L\otimes_AM)$ of Proposition 1.1 is an isomorphism. The graded $H_*(A)$-module $H_*(M)\cong H_*(A)\otimes_{\mathbb Z}H_*(S)$ is flat.

Proof. $L\otimes_AM=L\otimes_{\mathbb Z}S$, degreewise. By Eilenberg–Zilber the shuffle maps give quasi-isomorphisms $\operatorname{Tot}(NL\otimes NS)\to N(L\otimes S)$ and $\operatorname{Tot}(NA\otimes NS)\to N(A\otimes S)$. The complex $NS$ satisfies the hypotheses of Lemma 4.2, being a direct summand of the complex of $S$ with the same homology. So $H_*(L\otimes S)\cong H_*(L)\otimes H_*(S)$ and $H_*(M)\cong H_*(A)\otimes H_*(S)$. By associativity of shuffles, the map of Proposition 1.1 becomes the identity of $H_*(L)\otimes H_*(S)$ under $H_*(L)\otimes_{H_*(A)}(H_*(A)\otimes H_*(S))=$ $H_*(L)\otimes H_*(S)$. For a graded $H_*(A)$-module $V$, $V\otimes_{H_*(A)}H_*(M)=V\otimes_{\mathbb Z}H_*(S)$, which is exact in $V$ because every $H_n(S)$ is flat. $\square$

Lemma 4.4 (special resolutions). For every $A$-module $M$ there is an exact complex of $A$-modules $\cdots\to L_1(M)\to L_0(M)\to M\to0$, functorial in $M$, with special terms, such that $\cdots\to$ $H_*(L_1(M))\to$ $H_*(L_0(M))\to$ $H_*(M)\to$ $0$ is a resolution of $H_*(M)$ by flat graded $H_*(A)$-modules.

Proof. For a complex $C$ of abelian sheaves in degrees $\geq0$ and an abelian sheaf $V$, write $\mathbb Z^{(V)}$ for the free abelian sheaf on the underlying sheaf of sets, $S^n(V)$ for $V$ placed in degree $n$, and $D^n(V)$, $n\geq1$, for $V\xrightarrow{1}V$ in degrees $n$ and $n-1$. Put $\begin{aligned}&\Pi(C)\\&\quad =\bigoplus_{n\geq0}S^n\bigl(\mathbb Z^{(Z_nC)}\bigr)\oplus\bigoplus_{n\geq1}D^n\bigl(\mathbb Z^{(C_n)}\bigr),\end{aligned}$ mapping $S^n$ to $C$ through $\mathbb Z^{(Z_nC)}\to Z_nC\subseteq C_n$, and $D^n$ through $\mathbb Z^{(C_n)}\to C_n$ and its composite with the differential. This is a chain map, functorial in $C$, surjective in every degree (by $S^0$ in degree $0$ and by $D^n$ in degree $n\geq1$). Its terms are flat, and its homology is $\mathbb Z^{(Z_nC)}$ in degree $n$, which is flat and maps onto $H_n(C)$. Let $T(M)$ be the simplicial abelian sheaf corresponding to $\Pi(NM)$ under Dold–Kan. It has flat terms, being a finite sum of copies of the terms of $\Pi(NM)$ in each degree, and flat homology. The map $L_0(M)=A\otimes_{\mathbb Z}T(M)\to A\otimes_{\mathbb Z}M\to M$ is surjective, and surjective on homology: its composite with $T(M)\to A\otimes T(M)$, $t\mapsto1\otimes t$, is the map $T(M)\to M$. So $L_0(M)$ is special. Put $K_0=M$, $L_i(M)=L_0(K_i)$ and $K_{i+1}=\ker(L_i(M)\to K_i)$. Since each $L_i(M)\to K_i$ is surjective on homology, the long exact sequences break into short exact sequences $0\to$ $H_*(K_{i+1})\to$ $H_*(L_i(M))\to$ $H_*(K_i)\to$ $0$, and splicing gives the resolution of $H_*(M)$. Its terms are flat by Lemma 4.3. $\square$

Lemma 4.5 (filtered differential graded modules). Let $R$ be a differential graded ring of the topos, concentrated in degrees $\geq0$, and $M$ a differential graded $R$-module with an exhaustive increasing filtration by differential graded submodules $F_pM$, $p\geq0$, and $F_{-1}M=0$. From the first page on, the pages of the spectral sequence of the filtered complex $M$ are graded modules over the graded ring $H_*(R)$, with $d^r(ax)=(-1)^{|a|}a\,d^r(x)$, and each page is the homology of the previous one as an $H_*(R)$-module. The filtration of $H_*(M)$ by the images of the $H_*(F_pM)$ is a filtration by $H_*(R)$-submodules. When the spectral sequence converges, the isomorphisms of the limit page with the graded pieces of this filtration are $H_*(R)$-linear.

Proof. In homological indexing the page $E^r_p$ is $Z^r_p/(Z^{r-1}_{p-1}+B^{r-1}_p)$, where $Z^r_p$ consists of the sections $x$ of $F_pM$ with $dx$ in $F_{p-r}M$, and $B^{r-1}_p=F_pM\cap d(F_{p+r-1}M)$; the differential $d^r$ is induced by $d$ [Stacks, Tag 012M]. Let $a$ be a local section of $R$ of degree $m$ with $da=0$. Then $d(ax)=(-1)^ma\,dx$, and multiplication by $a$ preserves the filtration, so it maps $Z^r_p$, $Z^{r-1}_{p-1}$ and $B^{r-1}_p$ into themselves. It therefore acts on the pages, with $d^r(ax)=(-1)^ma\,d^r(x)$, compatibly with the passage from a page to the next. If $a=db$, then $ax=d(bx)+(-1)^mb\,dx$. For $x$ in $Z^r_p$ with $r\geq1$, $d(bx)$ lies in $F_pM\cap d(F_pM)\subseteq B^{r-1}_p$, and $b\,dx$ lies in $F_{p-r}M\subseteq F_{p-1}M$ with $d(b\,dx)=a\,dx$ in $F_{p-r}M$, so it lies in $Z^{r-1}_{p-1}$. Hence $a$ acts by zero, and the action factors through $H_*(R)$; it is associative and unital because the action of $R$ on $M$ is. A cycle $x$ of $F_pM$ gives the cycle $ax$ of $F_pM$, so the images of the $H_*(F_pM)$ are submodules. The isomorphism of the limit page with the graded pieces sends the class of a cycle of $F_pM$ to its class [Stacks, Tag 012M], so it commutes with multiplication by $a$. $\square$

Theorem 4.6 (Künneth spectral sequence). There is a spectral sequence, functorial in $E$ and $F$, $\begin{aligned}&E^2_{p,q} =\operatorname{Tor}^{H_*(A)}_p\bigl(H_*(E),H_*(F)\bigr)_q\\&\quad \Longrightarrow H_{p+q}(E\otimes^\ell_AF).\end{aligned}$ From the first page on it is a spectral sequence of graded $H_*(A)$-modules: every page is a graded $H_*(A)$-module, $d^r(ax)=(-1)^{|a|}a\,d^r(x)$, and the abutment is filtered by $H_*(A)$-submodules whose graded pieces are the limit page, as $H_*(A)$-modules. A quasi-isomorphism in either variable induces an isomorphism from the second page on.

Proof. Apply Dold–Kan to the resolution $L_\bullet(F)$ of Lemma 4.4 in the external direction. This gives a bisimplicial module $R$ with an augmentation $R\to cF$, and its column $p$ resolves $F_p$, since $L_\bullet(F)\to F$ is exact. The terms of $R$ are finite sums of terms of the $L_i(F)$, which are flat, so $\Delta R$ is flat, and $\Delta R\to F$ is a quasi-isomorphism by Lemma 1.1 of the lesson on derived tensor products. Hence $E\otimes^\ell_AF\cong E\otimes_A\Delta R=\Delta(cE\otimes_AR)$. By Eilenberg–Zilber, its homology is that of the total complex of the double complex with terms $N_k(E\otimes_AL_p(F))$, with $k$ the internal and $p$ the resolution degree. Consider the spectral sequence that first takes homology in the internal direction [Stacks, Tag 0132]. Its first page is $H_q(E\otimes_AL_p(F))$, which is $\bigl(H_*(E)\otimes_{H_*(A)}H_*(L_p(F))\bigr)_q$ by Lemma 4.3, with the differential induced by $L_\bullet(F)$, by naturality of the shuffle map. By Lemma 4.4, $H_*(L_\bullet(F))$ is a flat resolution of $H_*(F)$, so the second page is $\operatorname{Tor}^{H_*(A)}_p(H_*(E),H_*(F))_q$. The double complex lies in the first quadrant, so the spectral sequence converges. The construction is functorial in $E$ and in $F$, because $L_\bullet$ is. The second page depends only on $H_*(E)$ and $H_*(F)$, which proves the last statement.

Linearity. Take the internal direction first in Lemma 2.6 of the lesson on multisimplicial objects, so that the total complex $K$ of the double complex has the differential $d_{\mathrm{int}}+(-1)^kd_{\mathrm{ext}}$ on $N_k(E\otimes_AL_p(F))$. The differential graded ring $NA$ acts on every $N(E\otimes_AL_p(F))$ by shuffle maps (Proposition 2.1 of the lesson on simplicial rings and modules). The maps induced by $L_\bullet(F)$ are $A$-linear, so they commute with this action, and with the sign $(-1)^k$ the action makes $K$ a differential graded $NA$-module; the filtration by the resolution degree $p$ is a filtration by differential graded submodules. The shuffle map $\operatorname{sh}\colon K\to N\Delta(cE\otimes_AR)$ is $NA$-linear. Indeed, for $a$ of degree $m$ and $x$ of internal degree $k$ and external degree $p$, both $a\cdot\operatorname{sh}(x)$ and $\operatorname{sh}(a\cdot x)$ are the sum, over the shuffles of three lists of lengths $m$, $k$ and $p$, of the products of the degeneracies of $a$ with the internal and external degeneracies of $x$ that the shuffle prescribes, with its sign, as in the associativity of shuffle maps (Theorem 1.3 of The cotangent complex). Here the external degeneracies commute with the action, $A$ being constant in that direction. The isomorphism $E\otimes^\ell_AF\cong\Delta(cE\otimes_AR)$ is one of $A$-modules, so the identification of the abutment with $H_*(E\otimes^\ell_AF)$ is $H_*(A)$-linear. By Lemma 4.5, the spectral sequence is one of graded $H_*(A)$-modules from the first page on. On the first page the action is that of $H_*(A)$ on $H_*(E\otimes_AL_p(F))$, and the isomorphism with $H_*(E)\otimes_{H_*(A)}H_*(L_p(F))$ is $H_*(A)$-linear by Proposition 1.1, so the second page is $\operatorname{Tor}^{H_*(A)}_p(H_*(E),H_*(F))_q$ as an $H_*(A)$-module. $\square$

For a ring $\Lambda$ placed in every degree, $H_*(A)=\Lambda$ in degree $0$, and the second page is $\operatorname{Tor}^\Lambda_p(H_*(E),H_*(F))_q$, the second page of the Künneth spectral sequence of the complexes $NE$ and $NF$.

5. Exercises

5.1. Let $A=\mathbb Z$ in the topos of sets, and let $E$ and $F$ be simplicial abelian groups whose homology groups are free. Show that $H_*(E\otimes^\ell F)\cong H_*(E)\otimes H_*(F)$.

5.2. In the situation of Remark 2.2, compute the differentials of the two spectral sequences of Theorem 2.1 and check that both converge to $\mathbb Z/2$ in degrees $0$ and $1$.

5.3. Let $G$ be an $A$-module with $\pi_n(G)=0$ for $n\neq0$. Show that $G\otimes^\ell_AF\cong\pi_0(G)\otimes^\ell_AF$, and deduce from Theorem 3.4 that the Postnikov spectral sequence of $G\otimes^\ell_AF$ is concentrated in the row $q=0$.

6. Complete solutions

5.1. $H_*(A)=\mathbb Z$. The graded abelian group $H_*(F)$ is free, hence flat, so $\operatorname{Tor}^{\mathbb Z}_p(H_*(E),H_*(F))=0$ for $p>0$. The spectral sequence of Theorem 4.6 is concentrated in the column $p=0$, so it degenerates, and $H_n(E\otimes^\ell F)\cong(H_*(E)\otimes H_*(F))_n$.

5.2. For $E$, the second page is concentrated in the row $q=0$, so all differentials vanish and the abutment is $\mathbb Z/2$ in degrees $0$ and $1$. For the constant module, the second page is concentrated in the column $p=0$, and the differentials $d^r$, $r\geq2$, change $p$ by $-r$, so they vanish too. The abutment is $\mathbb Z/2$ in degrees $0=0+0$ and $1=0+1$. In both cases this is $\operatorname{Tor}^{\mathbb Z}_*(\mathbb Z/2,\mathbb Z/2)$.

5.3. By Lemma 3.3 with $q=0$, $G\cong\pi_0(G)$ in $D_\bullet(A)$, so $G\otimes^\ell_AF\cong\pi_0(G)\otimes^\ell_AF$. In Theorem 3.4 for $G$, $\pi_q(G)=0$ for $q\neq0$, so the second page is $H_p(\pi_0(G)\otimes^\ell_AF)$ in the row $q=0$ and zero elsewhere.

Sources

The three spectral sequences for the homology of derived tensor products of simplicial modules are due to Quillen, in his work on homotopical algebra, and to Illusie, in his work on the cotangent complex, where the Künneth spectral sequence over the homology ring is constructed with special modules. Its linearity over the homology ring is proved here through the shuffle maps, which make the double complex a filtered differential graded module over the normalized complex of $A$. The filtration of Definition 3.1 is Eilenberg's subcomplex of simplices with trivial low-dimensional faces. The behaviour of the first spectral sequence under quasi-isomorphisms is shown by the example of Remark 2.2.