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

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

2. 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.

Proof. Proof of (1). Suppose $K$ is given by the complex of modules $M^\bullet$. We may first replace $M^\bullet$ by $\tau _{\leq 0}M^\bullet$. Thus we may assume $M^ i = 0$ for $i > 0$, Next, we may choose a free resolution $P^\bullet \to M^\bullet$ with $P^ i = 0$ for $i > 0$, see Derived Categories, Lemma 13.15.4. Finally, we can set $K^\bullet = \tau _{\geq -1}P^\bullet$.

Proof of (2). Assume $R$ is Noetherian and $H^ i(K)$ is a finite $R$-module for $i = -1, 0$. By Lemma 15.64.5 we can choose a quasi-isomorphism $F^\bullet \to M^\bullet$ with $F^ i = 0$ for $i > 0$ and $F^ i$ finite free. Then we can set $K^\bullet = \tau _{\geq -1}F^\bullet$. $\square$

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