The Stacks project

Lemma 15.69.2. Let $K^\bullet $ be an object of $D(R)$. The following are equivalent

  1. $K^\bullet $ is perfect, and

  2. $K^\bullet $ is pseudo-coherent and has finite tor dimension.

If (1) and (2) hold and $K^\bullet $ has tor-amplitude in $[a, b]$, then $K^\bullet $ is quasi-isomorphic to a complex $E^\bullet $ of finite projective $R$-modules with $E^ i = 0$ for $i \not\in [a, b]$.

Proof. It is clear that (1) implies (2), see Lemmas 15.62.5 and 15.63.3. Assume (2) holds and that $K^\bullet $ has tor-amplitude in $[a, b]$. In particular, $H^ i(K^\bullet ) = 0$ for $i > b$. Choose a complex $F^\bullet $ of finite free $R$-modules with $F^ i = 0$ for $i > b$ and a quasi-isomorphism $F^\bullet \to K^\bullet $ (Lemma 15.62.5). Set $E^\bullet = \tau _{\geq a}F^\bullet $. Note that $E^ i$ is finite free except $E^ a$ which is a finitely presented $R$-module. By Lemma 15.63.2 $E^ a$ is flat. Hence by Algebra, Lemma 10.77.2 we see that $E^ a$ is finite projective. $\square$


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