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 $K^\bullet \in D^ b(R)$.
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. The second statement follows from Lemma 15.63.9 and the definitions. The third statement is a rephrasing of the second.
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