The Stacks project

Definition 21.46.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a, b \in \mathbf{Z}$ with $a \leq b$.

  1. We say $E$ has tor-amplitude in $[a, b]$ if $H^ i(E \otimes _\mathcal {O}^\mathbf {L} \mathcal{F}) = 0$ for all $\mathcal{O}$-modules $\mathcal{F}$ and all $i \not\in [a, b]$.

  2. We say $E$ has finite tor dimension if it has tor-amplitude in $[a, b]$ for some $a, b$.

  3. We say $E$ locally has finite tor dimension if for any object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} $ such that $E|_{U_ i}$ has finite tor dimension for all $i$.

An $\mathcal{O}$-module $\mathcal{F}$ has tor dimension $\leq d$ if $\mathcal{F}[0]$ viewed as an object of $D(\mathcal{O})$ has tor-amplitude in $[-d, 0]$.

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