Definition 15.65.1. Let $R$ be a ring. Denote $D(R)$ its derived category. Let $a, b \in \mathbf{Z}$.

An object $K^\bullet $ of $D(R)$ has

*tor-amplitude in $[a, b]$*if $H^ i(K^\bullet \otimes _ R^\mathbf {L} M) = 0$ for all $R$-modules $M$ and all $i \not\in [a, b]$.An object $K^\bullet $ of $D(R)$ has

*finite tor dimension*if it has tor-amplitude in $[a, b]$ for some $a, b$.An $R$-module $M$ has

*tor dimension $\leq d$*if $M[0]$ as an object of $D(R)$ has tor-amplitude in $[-d, 0]$.An $R$-module $M$ has

*finite tor dimension*if $M[0]$ as an object of $D(R)$ has finite tor dimension.

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