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.

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