Definition 15.66.1. Let $R$ be a ring. Denote $D(R)$ its derived category. Let $a, b \in \mathbf{Z}$.

1. 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 $i \not\in [a, b]$.

2. An object $K^\bullet$ of $D(R)$ has finite tor dimension if it has tor-amplitude in $[a, b]$ for some $a, b$.

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

4. 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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