The Stacks project

Lemma 15.65.9. Let $R$ be a ring. Let $a, b \in \mathbf{Z}$. Let $K^\bullet \in D^ b(R)$ such that $H^ i(K^\bullet )$ has tor amplitude in $[a - i, b - i]$ for all $i$. Then $K^\bullet $ has tor amplitude in $[a, b]$. In particular if $K^\bullet \in D^ b(R)$ and all its cohomology groups have finite tor dimension then $K^\bullet $ has finite tor dimension.

Proof. Follows by induction on the length of the finite complex: use Lemma 15.65.5 and the canonical truncations. $\square$


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