Lemma 97.23.1. 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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