Lemma 15.84.1 (0G9C)—The Stacks project

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Lemma 15.84.1. Let $R$ be a ring. Let $K \in D(R)$ with $H^ i(K) = 0$ for $i \not\in \{ -1, 0\}$ . The following are equivalent

$H^{-1}(K) = 0$ and $H^0(K)$ is a projective module and
$\mathop{\mathrm{Ext}}\nolimits ^1_ R(K, M) = 0$ for every $R$ -module $M$ .

If $R$ is Noetherian and $H^ i(K)$ is a finite $R$ -module for $i = -1, 0$ , then these are also equivalent to

$\mathop{\mathrm{Ext}}\nolimits ^1_ R(K, M) = 0$ for every finite $R$ -module $M$ .

Proof. The equivalence of (1) and (2) follows from Lemma 15.68.2. If $R$ is Noetherian and $H^ i(K)$ is a finite $R$ -module for $i = -1, 0$ , then $K$ is pseudo-coherent, see Lemma 15.64.17. Thus the equivalence of (1) and (3) follows from Lemma 15.77.4. $\square$

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