Lemma 15.84.3. Let $R$ be a ring. Let $K$ be an object of $D(R)$ with $H^ i(K) = 0$ for $i \not\in \{ -1, 0\} $. Then

$K$ can be represented by a two term complex $K^{-1} \to K^0$ with $K^0$ a free module, and

if $R$ is Noetherian and $H^ i(K)$ is a finite $R$-module for $i = -1, 0$, then $K$ can be represented by a two term complex $K^{-1} \to K^0$ with $K^0$ a finite free module and $K^{-1}$ finite.

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