Lemma 20.48.8 (08CK)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 20: Cohomology of Sheaves
Section 20.48: Tor dimension
Lemma 20.48.8 (cite)

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Lemma 20.48.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $a, b \in \mathbf{Z}$ . For $K$ , $L$ objects of $D(\mathcal{O}_ X)$ if $K \oplus L$ has tor amplitude in $[a, b]$ so do $K$ and $L$ .

Proof. Clear from the fact that the Tor functors are additive. $\square$

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