Lemma 21.44.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\mathcal{O})$. If $K$ has tor-amplitude in $[a, b]$ and $L$ has tor-amplitude in $[c, d]$ then $K \otimes _\mathcal {O}^\mathbf {L} L$ has tor amplitude in $[a + c, b + d]$.

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

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