Lemma 15.63.19. Let $R$ be a ring of finite global dimension $d$. Then

every module has tor dimension $\leq d$,

a complex of $R$-modules $K^\bullet $ with $H^ i(K^\bullet ) \not= 0$ only if $i \in [a, b]$ has tor amplitude in $[a - d, b]$, and

a complex of $R$-modules $K^\bullet $ has finite tor dimension if and only if $K^\bullet \in D^ b(R)$.

## Comments (2)

Comment #4012 by jojo on

Comment #4122 by Johan on

There are also: