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 $M$.

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