The Stacks project

Lemma 15.66.8. Let $R$ be a ring. Let $K^\bullet $ be a bounded complex of $R$-modules such that $K^ i$ has tor amplitude in $[a - i, b - i]$ for all $i$. Then $K^\bullet $ has tor amplitude in $[a, b]$. In particular if $K^\bullet $ is a finite complex of $R$-modules of finite tor dimension, then $K^\bullet $ has finite tor dimension.

Proof. Follows by induction on the length of the finite complex: use Lemma 15.66.5 and the stupid truncations. $\square$

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