Lemma 20.45.8. 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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