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

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

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.