Lemma 98.23.1 (07YK)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 98: Artin's Axioms
Section 98.23: Naive obstruction theories
Lemma 98.23.1 (cite)

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Lemma 98.23.1. Let $A \to k$ be a ring map with $k$ a field. Let $E \in D^-(A)$ . Then $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(E, k) = \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(E \otimes ^\mathbf {L} k), k)$ .

Proof. Omitted. Hint: Replace $E$ by a bounded above complex of free $A$ -modules and compute both sides. $\square$

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