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
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.
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