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The Stacks project

Lemma 48.27.4. Let k, X, and \omega _ X^\bullet be as in Lemma 48.27.1. Let t : H^0(X, \omega _ X^\bullet ) \to k be as in Remark 48.27.2. Let E \in D(\mathcal{O}_ X) be perfect. Then the pairings

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

are perfect for all i. Here \cup denotes the cupproduct of Cohomology, Section 20.31 and \epsilon : E^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} E \to \mathcal{O}_ X is as in Cohomology, Example 20.50.7.

Proof. By replacing E with E[-i] this reduces to the case i = 0. By Cohomology, Lemma 20.51.2 we see that the pairing is the same as the one discussed in Remark 48.27.2 whence the result by the discussion in that remark. \square


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