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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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