Lemma 48.27.4 (0FVY)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 48: Duality for Schemes
Section 48.27: Duality for proper schemes over fields
Lemma 48.27.4 (cite)

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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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