The Stacks project

Remark 48.27.6. Let $X$ be a proper Cohen-Macaulay scheme over a field $k$ which is equidimensional of dimension $d$. Let $\omega _ X^\bullet $ and $\omega _ X$ be as in Lemma 48.27.1. By Lemma 48.27.5 we have $\omega _ X^\bullet = \omega _ X[d]$. Let $t : H^ d(X, \omega _ X) \to k$ be the map of Remark 48.27.2. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module with dual $\mathcal{E}^\vee $. Then we have perfect pairings

\[ H^ i(X, \omega _ X \otimes _{\mathcal{O}_ X} \mathcal{E}^\vee ) \times H^{d - i}(X, \mathcal{E}) \longrightarrow k,\quad (\xi , \eta ) \longmapsto t(1 \otimes \epsilon )(\xi \cup \eta )) \]

where $\cup $ is the cup-product and $\epsilon : \mathcal{E}^\vee \otimes _{\mathcal{O}_ X} \mathcal{E} \to \mathcal{O}_ X$ is the evaluation map. This is a special case of Lemma 48.27.4.

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