Definition 20.45.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_ X)$. Let $a, b \in \mathbf{Z}$ with $a \leq b$.

We say $E$ has

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

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

*locally has finite tor dimension*if there exists an open covering $X = \bigcup U_ i$ such that $E|_{U_ i}$ has finite tor dimension for all $i$.

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

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