The Stacks project

Remark 48.27.3. We continue the discussion in Remark 48.27.2 and we use the same notation $k$, $X$, $\omega _ X^\bullet $, and $t$. If $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module we obtain perfect pairings

\[ \langle -, - \rangle : \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{F}, \omega _ X^\bullet ) \times H^{-i}(X,\mathcal{F}) \longrightarrow k,\quad (\alpha , \beta ) \longmapsto t(\alpha (\beta )) \]

of finite dimensional $k$-vector spaces. These pairings satisfy the following (obvious) functoriality: if $\varphi : \mathcal{F} \to \mathcal{G}$ is a homomorphism of coherent $\mathcal{O}_ X$-modules, then we have

\[ \langle \alpha \circ \varphi , \beta \rangle = \langle \alpha , \varphi (\beta ) \rangle \]

for $\alpha \in \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{G}, \omega _ X^\bullet )$ and $\beta \in H^{-i}(X, \mathcal{F})$. In other words, the $k$-linear map $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{G}, \omega _ X^\bullet ) \to \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{F}, \omega _ X^\bullet )$ induced by $\varphi $ is, via the pairings, the $k$-linear dual of the $k$-linear map $H^{-i}(X, \mathcal{F}) \to H^{-i}(X, \mathcal{G})$ induced by $\varphi $. Formulated in this manner, this still works if $\varphi $ is a homomorphism of quasi-coherent $\mathcal{O}_ X$-modules.


Comments (0)


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Your email address will not be published.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