The Stacks project

Lemma 75.13.3. Let $X$ be a scheme. Let $E$ be an object of $D(\mathcal{O}_ X)$. Then

  1. $E$ has tor amplitude in $[a, b]$ if and only if $\epsilon ^*E$ has tor amplitude in $[a, b]$.

  2. $E$ has finite tor dimension if and only if $\epsilon ^*E$ has finite tor dimension.

Here $\epsilon $ is as in (75.4.0.1).

Proof. The easy implication follows from Cohomology on Sites, Lemma 21.46.5. For the converse, assume that $\epsilon ^*E$ has tor amplitude in $[a, b]$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. As $\epsilon $ is a flat morphism of ringed sites (Lemma 75.4.1) we have

\[ \epsilon ^*(E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{F}) = \epsilon ^*E \otimes ^\mathbf {L}_{\mathcal{O}_{\acute{e}tale}} \epsilon ^*\mathcal{F} \]

Thus the (assumed) vanishing of cohomology sheaves on the right hand side implies the desired vanishing of the cohomology sheaves of $E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{F}$ via Lemma 75.4.1. $\square$

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