The Stacks project

35.22 Künneth formula

For the case where the base is a field, please see Varieties, Section 32.29. Consider a cartesian diagram of schemes

\[ \xymatrix{ & X \times _ S Y \ar[ld]^ p \ar[rd]_ q \ar[dd]^ f \\ X \ar[rd]_ a & & Y \ar[ld]^ b \\ & S } \]

Let $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ and $M \in D_\mathit{QCoh}(\mathcal{O}_ Y)$. There is a canonical map
\begin{equation} \label{perfect-equation-kunneth} Ra_*K \otimes _{\mathcal{O}_ S}^\mathbf {L} Rb_*M \longrightarrow Rf_*(Lp^*K \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} Lq^*M) \end{equation}

Namely, we can use the maps $Ra_*K \to Ra_*Rp_* Lp^*K = Rf_*Lp^*K$ and $Rb_*M \to Rb_*Rq_* Lq^*M = Rf_*Lq^*M$ and then we can use the relative cup product (Cohomology, Remark 20.28.6).

Lemma 35.22.1. In the situation above, if $a$ and $b$ are quasi-compact and quasi-separated and $X$ and $Y$ are tor-independent over $S$, then ( is an isomorphism.

Proof. This follows from the following sequence of isomorphisms

\begin{align*} Rf_*(Lp^*K \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} Lq^*M) & = Ra_*Rp_*(Lp^*K \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} Lq^*M) \\ & = Ra_*(K \otimes _{\mathcal{O}_{X}}^\mathbf {L} Rp_*Lq^*M) \\ & = Ra_*(K \otimes _{\mathcal{O}_{X}}^\mathbf {L} La^*Rb_*M) \\ & = Ra_*K \otimes _{\mathcal{O}_{X}}^\mathbf {L} Rb_*M \end{align*}

The first equality holds because $f = a \circ p$. The second equality by Lemma 35.21.1. The third equality by Lemma 35.21.3. The fourth equality by Lemma 35.21.1. We omit the verification that the composition of these isomorphisms is the same as the map ( $\square$

In the situation above assume $S = \mathop{\mathrm{Spec}}(A)$ affine. Let $\mathcal{F}^\bullet $ be a bounded below complex of $a^{-1}\mathcal{O}_ S$-modules. Let $\mathcal{G}^\bullet $ be a bounded below complex of $b^{-1}\mathcal{O}_ S$-modules. Then there exists a “cup product” map
\begin{equation} \label{perfect-equation-de-rham-kunneth} R\Gamma (X, \mathcal{F}^\bullet ) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G}^\bullet ) \longrightarrow R\Gamma (X \times _ S Y, \text{Tot}(p^{-1}\mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet )) \end{equation}

in $D(A)$. This is constructed using the pullback maps $R\Gamma (X, \mathcal{F}^\bullet ) \to R\Gamma (X \times _ S Y, p^{-1}\mathcal{F}^\bullet )$ and $R\Gamma (Y, \mathcal{G}^\bullet ) \to R\Gamma (X \times _ S Y, q^{-1}\mathcal{G}^\bullet )$ and the cup product constructed in Cohomology, Section 20.31.

Remark 35.22.2. In the situation above suppose we have an $\mathcal{O}_ X$-module $\mathcal{F}$ and an $\mathcal{O}_ Y$-module $\mathcal{G}$. Then we have

\begin{align*} & p^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G} \\ & = p^{-1}(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X) \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}(\mathcal{O}_ Y \otimes _{\mathcal{O}_ Y} \mathcal{G}) \\ & = p^{-1}\mathcal{F} \otimes _{p^{-1}\mathcal{O}_ X} p^{-1}\mathcal{O}_ X \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{O}_ Y \otimes _{q^{-1}\mathcal{O}_ Y} q^{-1}\mathcal{G} \\ & = p^{-1}\mathcal{F} \otimes _{q^{-1}\mathcal{O}_ X} \mathcal{O}_{X \times _ S Y} \otimes _{q^{-1}\mathcal{O}_ Y} q^{-1}\mathcal{G} \\ & = p^{-1}\mathcal{F} \otimes _{q^{-1}\mathcal{O}_ X} \mathcal{O}_{X \times _ S Y} \otimes _{\mathcal{O}_{X \times _ S Y}} \mathcal{O}_{X \times _ S Y} \otimes _{q^{-1}\mathcal{O}_ Y} q^{-1}\mathcal{G} \\ & = p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G} \end{align*}

Lemma 35.22.3. In the situation above assume

  1. $X$ and $Y$ quasi-compact with affine diagonal,

  2. $\mathcal{F}^ n$ is a flat $p^{-1}\mathcal{O}_ S$-module and $\mathcal{F}^ n$ has the structure of a quasi-coherent $\mathcal{O}_ X$-module compatible with the given $p^{-1}\mathcal{O}_ S$-module structure (but the differentials in the complex $\mathcal{F}^\bullet $ need not be $\mathcal{O}_ X$-linear),

  3. $\mathcal{G}^ m$ is a flat $q^{-1}\mathcal{O}_ S$-module and $\mathcal{G}^ m$ has the structure of a quasi-coherent $\mathcal{O}_ Y$-module compatible with the given $q^{-1}\mathcal{O}_ S$-module structure (but the differentials in the complex $\mathcal{G}^\bullet $ need not be $\mathcal{O}_ Y$-linear).

Then the cup product ( is an isomorphism in $D(A)$.

This lemma is true without the assumption on the diagonals of $X$ and $Y$. The assumption on the diagonals can be removed by replacing open coverings in the proof below by hypercoverings.

Proof. We will prove the result on the cup product in two steps: (1) we reduce to the case where each of the two complexes consist of a single sheaf placed in degree $0$ and (2) the case of a single sheaf is proved as in Varieties, Lemma 32.29.1.

Suppose that we have maps of complexes of $p^{-1}\mathcal{O}_ S$-modules

\[ \mathcal{F}_1^\bullet \to \mathcal{F}_2^\bullet \to \mathcal{F}_3^\bullet \to \mathcal{F}_1^\bullet [1] \]

Then by the functoriality of the cup product we obtain a commutative diagram

\[ \xymatrix{ R\Gamma (X, \mathcal{F}_1^\bullet ) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G}^\bullet ) \ar[r] \ar[d] & R\Gamma (X \times _ S Y, \text{Tot}(p^{-1}\mathcal{F}_1^\bullet \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet )) \ar[d] \\ R\Gamma (X, \mathcal{F}_2^\bullet ) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G}^\bullet ) \ar[r] \ar[d] & R\Gamma (X \times _ S Y, \text{Tot}(p^{-1}\mathcal{F}_2^\bullet \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet )) \ar[d] \\ R\Gamma (X, \mathcal{F}_3^\bullet ) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G}^\bullet ) \ar[r] \ar[d] & R\Gamma (X \times _ S Y, \text{Tot}(p^{-1}\mathcal{F}_3^\bullet \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet )) \ar[d] \\ R\Gamma (X, \mathcal{F}_1^\bullet [1]) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G}^\bullet ) \ar[r] & R\Gamma (X \times _ S Y, \text{Tot}(p^{-1}\mathcal{F}_1^\bullet [1] \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet )) } \]

If the original maps form a distinguished triangle in the homotopy category of complexes of $p^{-1}\mathcal{O}_ S$-modules, then the columns of this diagram form distinguished triangles in $D(A)$.

Suppose that $\mathcal{F}^ n = 0$ for $n < a$. Then we may consider the termwise split short exact sequence of complexes

\[ 0 \to \sigma _{\geq a + 1}\mathcal{F}^\bullet \to \mathcal{F}^\bullet \to \mathcal{F}^ a[-a] \to 0 \]

where the truncation is as in Homology, Section 12.15. This produces the distinguished triangle

\[ \sigma _{\geq a + 1}\mathcal{F}^\bullet \to \mathcal{F}^\bullet \to \mathcal{F}^ a[-a] \to (\sigma _{\geq a + 1}\mathcal{F}^\bullet )[1] \]

where the final arrow is given by the boundary map $\mathcal{F}^ a \to \mathcal{F}^{a + 1}$. It follows from the discussion above that it suffices to prove the lemma for $\mathcal{F}^ a[-a]$ and $\sigma _{\geq a + 1}\mathcal{F}^\bullet $. Hence if we can prove the lemma if $\mathcal{F}^\bullet $ is nonzero only in single degree, then the lemma follows. Namely, $R\Gamma (X, \sigma _{\geq N}\mathcal{F}^\bullet )$ has no nonzero cohomology in degree $< N$ and both $R\Gamma (X, \sigma _{\geq n}\mathcal{F}^\bullet ) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G}^\bullet )$ and $R\Gamma (X \times _ S Y, \text{Tot}(p^{-1}\sigma _{\geq N}\mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet ))$ have no nonzero cohomology in degree $< N + b$ if $\mathcal{G}^ m = 0$ for $m < b$. Thus we may assume $\mathcal{F}^\bullet $ is nonzero only in one degree. Repeating the argument with $\mathcal{G}^\bullet $ we reduce to the case discussed in the next paragraph.

Assume $\mathcal{F}^\bullet = \mathcal{F}$ and $\mathcal{G}^\bullet = \mathcal{G}$ are both reduced to a single quasi-coherent module placed in degree $0$ and both $\mathcal{F}$ and $\mathcal{G}$ are flat over $S$. Choose finite affine open coverings $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ and $\mathcal{V} : Y = \bigcup _{j \in J} V_ j$. This determines an affine open covering $\mathcal{W} : X \times _ S Y = \bigcup _{(i, j) \in I \times J} U_ i \times _ S V_ j$. Note that $\mathcal{W}$ is a refinement of $\text{pr}_1^{-1}\mathcal{U}$ and of $\text{pr}_2^{-1}\mathcal{V}$. Thus by the discussion in Cohomology, Section 20.25 we obtain maps

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F}) \quad \text{and}\quad \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G}) \to \check{\mathcal{C}}^\bullet (\mathcal{W}, q^*\mathcal{G}) \]

well defined up to homotopy and compatible with pullback maps on cohomology. In Cohomology, Equation ( we have constructed a map of complexes

\[ \text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F}) \otimes _ A \check{\mathcal{C}}^\bullet (\mathcal{W}, q^*\mathcal{G})) \longrightarrow \check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G}) \]

which is compatible with the cup product on cohomology by Cohomology, Lemma 20.31.1. Combining the above we obtain a map of complexes
\begin{equation} \label{perfect-equation-kunneth-on-cech} \text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G})) \to \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G})) \end{equation}

We claim this is the map in the statement of the lemma, i.e., the source and target of this arrow are the same as the source and target of ( Namely, by Cohomology of Schemes, Lemma 29.2.2 and Cohomology, Lemma 20.25.2 the canonical maps

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \to R\Gamma (X, \mathcal{F}), \quad \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G}) \to R\Gamma (Y, \mathcal{G}) \]


\[ \check{\mathcal{C}}^\bullet (\mathcal{W}, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G}) \to R\Gamma (X \times _ S Y, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G}) \]

are isomorphisms. On the other hand, the complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is homotopy equivalent to the alternating Čech complex $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ by Cohomology, Lemma 20.23.6. As the modules $\mathcal{F}^ n(U_{i_0\ldots i_ s})$ are $A$-flat, we see that the complex $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ is K-flat (More on Algebra, Lemma 15.57.9). Hence $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is a K-flat complex of $A$-modules too and we conclude that $\text{Tot}( \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G}))$ represents the derived tensor product $R\Gamma (X, \mathcal{F}) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G})$ as claimed. Finally, we have $p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G} = p^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}$ by Remark 35.22.2.

We still have to show that ( is a quasi-isomorphism. We will do this using dimension shifting. Set $d(\mathcal{F}) = \max \{ d \mid H^ d(X, \mathcal{F}) \not= 0\} $. Assume $d(\mathcal{F}) > 0$. Set $U = \coprod \nolimits _{i \in I} U_ i$. This is an affine scheme as $I$ is finite. Denote $j : U \to X$ the morphism which is the inclusion $U_ i \to X$ on each $U_ i$. Since the diagonal of $X$ is affine, the morphism $j$ is affine, see Morphisms, Lemma 28.11.11. It follows that $\mathcal{F}' = j_*j^*\mathcal{F}$ is $S$-flat, see Morphisms, Lemma 28.24.4. It also follows that $d(\mathcal{F}') = 0$ by combining Cohomology of Schemes, Lemmas 29.2.4 and 29.2.2. For all $x \in X$ we have $\mathcal{F}_ x \to \mathcal{F}'_ x$ is the inclusion of a direct summand: if $x \in U_ i$, then $\mathcal{F}' \to (U_ i \to X)_*\mathcal{F}|_{U_ i}$ gives a splitting. We conclude that $\mathcal{F} \to \mathcal{F}'$ is injective and $\mathcal{F}'' = \mathcal{F}'/\mathcal{F}$ is $S$-flat as well. Also $d(\mathcal{F}'') < d(\mathcal{F})$. In this way we reduce to the case $d(\mathcal{F}) = 0$.

Arguing in the same fashion for $\mathcal{G}$ we find that we may assume that both $\mathcal{F}$ and $\mathcal{G}$ have nonzero cohomology only in degree $0$. Observe that this means that $\Gamma (X, \mathcal{F})$ is quasi-isomorphic to the finite complex $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$ of flat $A$-modules sitting in degrees $0, \ldots , |I|$. It follows that $\Gamma (X, \mathcal{F})$ is a flat $A$-module. Let $V \subset Y$ be an affine open. Consider the affine open covering $\mathcal{U}_ V : X \times _ S V = \bigcup _{i \in I} U_ i \times _ S V$. It is immediate that

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A \mathcal{G}(V) = \check{\mathcal{C}}^\bullet (\mathcal{U}_ V, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G}) \]

(equality of complexes). By the flatness of $\mathcal{G}(V)$ over $A$ we see that $\Gamma (X, \mathcal{F}) \otimes _ A \mathcal{G}(V) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A \mathcal{G}(V)$ is a quasi-isomorphism. Since the sheafification of $V \mapsto \check{\mathcal{C}}^\bullet (\mathcal{U}_ V, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G})$ represents $Rq_*(p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G})$ by Cohomology of Schemes, Lemma 29.7.1 we conclude that

\[ Rq_*(p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G}) \cong \Gamma (X, \mathcal{F}) \otimes _ A \mathcal{G} \]

on $Y$ with obvious notation. Using this and Lemma 35.21.1 (which applies as $\Gamma (X, \mathcal{F})$ is $A$-flat) we conclude that $H^ n(X \times _ S Y, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G})$ is zero for $n > 0$ and isomorphic to $H^0(X, \mathcal{F}) \otimes _ A H^0(Y, \mathcal{G})$ for $n = 0$. This finishes the proof (except that we should check that the isomorphism is indeed given by cup product in degree $0$; we omit the verification). $\square$

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