Lemma 20.44.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L$ be objects of $D(\mathcal{O}_ X)$. If $K$ has tor-amplitude in $[a, b]$ and $L$ has tor-amplitude in $[c, d]$ then $K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ has tor amplitude in $[a + c, b + d]$.

**Proof.**
Omitted. Hint: use the spectral sequence for tors.
$\square$

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