Lemma 36.22.6. The cup product (36.22.4.1) is an isomorphism in $D(A)$ if the following assumptions hold

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

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

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

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

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

Proof. 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 $

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

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

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 $n < i$. 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

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

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 36.22.3 for example. Say it has tor amplitude in $[i, j]$. Pick $N \gg 0$ and consider the distinguished triangle

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

in the homotopy category of complexes of $b^{-1}\mathcal{O}_ S$-modules on $Y$. Now observe that both

$R\Gamma (X, \mathcal{F}) \otimes _ A^\mathbf {L} R\Gamma (Y, \sigma _{\geq N + 1}\mathcal{G}^\bullet ) \quad \text{and}\quad R\Gamma (X \times _ S Y, \mathcal{F} \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\sigma _{\geq N + 1}\mathcal{G}^\bullet ))$

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 36.22.5. $\square$

Comment #4650 by on

First, of all, we should add a variant of this lemma where $X$ and $Y$ are flat over $S$ in which case this lemma is a lot easier to prove. Secondly, in the formula $R\Gamma (X, \sigma _{\geq n}\mathcal{F}^\bullet ) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G}^\bullet )$ there is a typo and $n$ should be $N$. Thirdly, the statement that this lives in degrees $\geq N + b$ needs to be justified by proving a bound on the tor dimension of $R\Gamma (Y, \mathcal{G}^\bullet )$ which follows from the results in the next paragraph.

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