Lemma 15.65.19. Let $R$ be a ring of finite global dimension $d$. Then

1. every module has tor dimension $\leq d$,

2. 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

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

Proof. The assumption on $R$ means that every module has a finite projective resolution of length at most $d$, in particular every module has tor dimension $\leq d$. The second statement follows from Lemma 15.65.9 and the definitions. The third statement is a rephrasing of the second. $\square$

Comment #4012 by jojo on

"The assumption on $R$ means that every module has a finite projective resolution of length at most $d$, in particular every module has finite tor dimension."

you should say

""The assumption on $R$ means that every module has a finite projective resolution of length at most $d$, in particular every module has tor dimension less $\leq d$."

since this is what you are proving ?

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