The Stacks project

36.24 Künneth formula, III

Let $X, Y, S, a, b, p, q, f$ be as in the introduction to Section 36.23. In this section, given an $\mathcal{O}_ X$-module $\mathcal{F}$ and a $\mathcal{O}_ Y$-module $\mathcal{G}$ let us set

\[ \mathcal{F} \boxtimes \mathcal{G} = p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G} \]

Note that, contrary to what happens in a future section, we take the nonderived tensor product here.

On $X$ let $\mathcal{F}^\bullet $ be a complex of sheaves of abelian groups whose terms are quasi-coherent $\mathcal{O}_ X$-modules such that the differentials $d^ i_\mathcal {F} : \mathcal{F}^ i \to \mathcal{F}^{i + 1}$ are differential operators on $X/S$ of finite order, see Morphisms, Section 29.33. Similarly, on $Y$ let $\mathcal{G}^\bullet $ be a complex of sheaves of abelian groups whose terms are quasi-coherent $\mathcal{O}_ Y$-modules such that the differentials $d^ j_\mathcal {G} : \mathcal{G}^ j \to \mathcal{G}^{j + 1}$ are differential operators on $Y/S$ of finite order. Applying the construction of Morphisms, Lemma 29.33.2 we obtain a double complex

\[ \xymatrix{ \ldots & \ldots & \ldots & \ldots \\ \ldots \ar[r] & \mathcal{F}^ i \boxtimes \mathcal{G}^{j + 1} \ar[r]^{d_1^{i, j + 1}} \ar[u] & \mathcal{F}^{i + 1} \boxtimes \mathcal{G}^{j + 1} \ar[r] \ar[u] & \ldots \\ \ldots \ar[r] & \mathcal{F}^ i \boxtimes \mathcal{G}^ j \ar[r]^{d_1^{i, j}} \ar[u]^{d_2^{i, j}} & \mathcal{F}^{i + 1} \boxtimes \mathcal{G}^ j \ar[r] \ar[u]_{d_2^{i + 1, j}} & \ldots \\ \ldots & \ldots \ar[u] & \ldots \ar[u] & \ldots } \]

of quasi-coherent modules whose maps are differential operators of finite order on $X \times _ S Y / S$. Please see the discussion in Morphisms, Remark 29.33.3 and Homology, Example 12.18.2. To be explicit, we set

\[ d_1^{i, j} = d^ i_\mathcal {F} \boxtimes 1 \quad \text{and}\quad d_2^{i, j} = 1 \boxtimes d^ j_\mathcal {G} \]

In the discussion below the notation

\[ \text{Tot}(\mathcal{F}^\bullet \boxtimes \mathcal{G}^\bullet ) \]

refers to the total complex associated to this double complex. This complex has terms which are quasi-coherent $\mathcal{O}_{X \times _ S Y}$-modules and whose differentials are differential operators of finite order on $X \times _ S Y / S$.

In the situation above there exists a “relative cup product” map

36.24.0.1
\begin{equation} \label{perfect-equation-relative-de-rham-kunneth} Ra_*(\mathcal{F}^\bullet ) \otimes _{\mathcal{O}_ S}^\mathbf {L} Rb_*(\mathcal{G}^\bullet ) \longrightarrow Rf_*\left(\text{Tot}(\mathcal{F}^\bullet \boxtimes \mathcal{G}^\bullet )\right) \end{equation}

Namely, we can construct this map by combining

  1. $Ra_*(\mathcal{F}^\bullet ) \to Rf_*(p^{-1}\mathcal{F}^\bullet )$,

  2. $Rb_*(\mathcal{G}^\bullet ) \to Rf_*(q^{-1}\mathcal{G}^\bullet )$,

  3. $Rf_*(p^{-1}\mathcal{F}^\bullet ) \otimes _{\mathcal{O}_ S}^\mathbf {L} Rf_*(q^{-1}\mathcal{G}^\bullet ) \to Rf_*(p^{-1}\mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ S}^\mathbf {L} q^{-1}\mathcal{G}^\bullet )$,

  4. $p^{-1}\mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ S}^\mathbf {L} q^{-1}\mathcal{G}^\bullet \to \text{Tot}(p^{-1}\mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet )$

  5. $\text{Tot}(p^{-1}\mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet ) \to \text{Tot}(\mathcal{F}^\bullet \boxtimes \mathcal{G}^\bullet )$.

Maps (1) and (2) are pullback maps, map (3) is the relative cup product, see Cohomology, Remark 20.28.7, map (4) compares the derived and nonderived tensor products, and map (5) is given by the obvious maps $p^{-1}\mathcal{F}^ i \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^ j \to \mathcal{F}^ i \boxtimes \mathcal{G}^ j$ on the underlying double complexes.

Set $A = \Gamma (S, \mathcal{O}_ S)$. There exists a “global cup product” map

36.24.0.2
\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}(\mathcal{F}^\bullet \boxtimes \mathcal{G}^\bullet )) \end{equation}

in $D(A)$. This is constructed similarly to the relative cup product above using

  1. $R\Gamma (X, \mathcal{F}^\bullet ) \to R\Gamma (X \times _ S Y, p^{-1}\mathcal{F}^\bullet )$

  2. $R\Gamma (Y, \mathcal{G}^\bullet ) \to R\Gamma (X \times _ S Y, q^{-1}\mathcal{G}^\bullet )$,

  3. $R\Gamma (X \times _ S Y, p^{-1}\mathcal{F}^\bullet ) \otimes _ A^\mathbf {L} R\Gamma (X \times _ S Y, q^{-1}\mathcal{G}^\bullet ) \to R\Gamma (X \times _ S Y, p^{-1}\mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ S}^\mathbf {L} q^{-1}\mathcal{G}^\bullet )$,

  4. $p^{-1}\mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ S}^\mathbf {L} q^{-1}\mathcal{G}^\bullet \to \text{Tot}(p^{-1}\mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet )$

  5. $\text{Tot}(p^{-1}\mathcal{F}^\bullet \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G}^\bullet ) \to \text{Tot}(\mathcal{F}^\bullet \boxtimes \mathcal{G}^\bullet )$.

Here maps (1) and (2) are the pullback maps, map (3) is the cup product constructed in Cohomology, Section 20.31. Maps (4) and (5) are as indicated in the previous paragraph.

Lemma 36.24.1. In the situation above the cup product (36.24.0.2) is an isomorphism in $D(A)$ if the following assumptions hold

  1. $S = \mathop{\mathrm{Spec}}(A)$ is affine,

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

  3. $\mathcal{F}^\bullet $ is bounded,

  4. $\mathcal{G}^\bullet $ is bounded below,

  5. $\mathcal{F}^ n$ is $S$-flat, and

  6. $\mathcal{G}^ m$ is $S$-flat.

Proof. We will use the notation $\mathcal{A}_{X/S}$ and $\mathcal{A}_{Y/S}$ introduced in Morphisms, Remark 29.33.3. Suppose that we have maps of complexes

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

in the category $\mathcal{A}_{X/S}$. Then by the functoriality of the cup product we obtain a commutative diagram

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

If the original maps form a distinguished triangle in the homotopy category of $\mathcal{A}_{X/S}$, then the columns of this diagram form distinguished triangles in $D(A)$.

In the situation of the lemma, suppose that $\mathcal{F}^ n = 0$ for $n < i$. Then we may consider the termwise split short exact sequence of complexes

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

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

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

in the homotopy category of $\mathcal{A}_{X/S}$ where the final arrow is given by the boundary map $\mathcal{F}^ i \to \mathcal{F}^{i + 1}$. It follows from the discussion above that it suffices to prove the lemma for $\mathcal{F}^ i[-i]$ and $\sigma _{\geq i + 1}\mathcal{F}^\bullet $. Since $\sigma _{\geq i + 1}\mathcal{F}^\bullet $ has fewer nonzero terms, by induction, if we can prove the lemma if $\mathcal{F}^\bullet $ is nonzero only in single degree, then the lemma follows. Thus we may assume $\mathcal{F}^\bullet $ is nonzero only in one degree.

Assume $\mathcal{F}^\bullet $ is the complex which has an $S$-flat quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ sitting in degree $0$ and is zero in other degrees. Observe that $R\Gamma (X, \mathcal{F})$ has finite tor dimension by Lemma 36.23.2 for example. Say it has tor amplitude in $[i, j]$. Pick $N \gg 0$ and consider the distinguished triangle

\[ \sigma _{\geq N + 1}\mathcal{G}^\bullet \to \mathcal{G}^\bullet \to \sigma _{\leq N}\mathcal{G}^\bullet \to (\sigma _{\geq N + 1}\mathcal{G}^\bullet )[1] \]

in the homotopy category of $\mathcal{A}_{Y/S}$. Now observe that both

\[ R\Gamma (X, \mathcal{F}) \otimes _ A^\mathbf {L} R\Gamma (Y, \sigma _{\geq N + 1}\mathcal{G}^\bullet ) \quad \text{and}\quad R\Gamma (X \times _ S Y, \text{Tot}(\mathcal{F} \boxtimes \sigma _{\geq N + 1}\mathcal{G}^\bullet )) \]

have vanishing cohomology in degrees $\leq N + i$. Thus, using the arguments given above, if we want to prove our statement in a given degree, then we may assume $\mathcal{G}^\bullet $ is bounded. Repeating the arguments above one more time we may also assume $\mathcal{G}^\bullet $ is nonzero only in one degree. This case is handled by Lemma 36.23.4. $\square$

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