The Stacks project

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

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

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 $n < i$. 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

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

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

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

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. $\square$


Comments (2)

Comment #4650 by on

First, of all, we should add a variant of this lemma where and are flat over in which case this lemma is a lot easier to prove. Secondly, in the formula there is a typo and should be . Thirdly, the statement that this lives in degrees needs to be justified by proving a bound on the tor dimension of which follows from the results in the next paragraph.


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FLT. Beware of the difference between the letter 'O' and the digit '0'.