The Stacks project

Lemma 36.12.1. Let $f : X \to Y$ be a surjective flat morphism of schemes (or more generally locally ringed spaces). Let $E \in D(\mathcal{O}_ Y)$. Let $a, b \in \mathbf{Z}$. Then $E$ has tor-amplitude in $[a, b]$ if and only if $Lf^*E$ has tor-amplitude in $[a, b]$.

Proof. Pullback always preserves tor-amplitude, see Cohomology, Lemma 20.48.4. We may check tor-amplitude in $[a, b]$ on stalks, see Cohomology, Lemma 20.48.5. A flat local ring homomorphism is faithfully flat by Algebra, Lemma 10.39.17. Thus the result follows from More on Algebra, Lemma 15.66.17. $\square$

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