The Stacks project

Remark 20.28.7. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The adjointness of $Lf^*$ and $Rf_*$ allows us to construct a relative cup product

\[ Rf_*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L \longrightarrow Rf_*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \]

in $D(\mathcal{O}_ Y)$ for all $K, L$ in $D(\mathcal{O}_ X)$. Namely, this map is adjoint to a map $Lf^*(Rf_*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L) \to K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ for which we can take the composition of the isomorphism $Lf^*(Rf_*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L) = Lf^*Rf_*K \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*Rf_*L$ (Lemma 20.27.3) with the map $Lf^*Rf_*K \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*Rf_*L \to K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ coming from the counit $Lf^* \circ Rf_* \to \text{id}$.


Comments (0)

There are also:

  • 2 comment(s) on Section 20.28: Cohomology of unbounded complexes

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Your email address will not be published.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 0B68. Beware of the difference between the letter 'O' and the digit '0'.