Lemma 15.66.13. Let $A \to B$ be a ring map. Let $a, b \in \mathbf{Z}$. Let $K^\bullet $ be a complex of $A$-modules with tor amplitude in $[a, b]$. Then $K^\bullet \otimes _ A^{\mathbf{L}} B$ as a complex of $B$-modules has tor amplitude in $[a, b]$.

**Proof.**
By Lemma 15.66.3 we can find a quasi-isomorphism $E^\bullet \to K^\bullet $ where $E^\bullet $ is a complex of flat $A$-modules with $E^ i = 0$ for $i \not\in [a, b]$. Then $E^\bullet \otimes _ A B$ computes $K^\bullet \otimes _ A ^{\mathbf{L}} B$ by construction and each $E^ i \otimes _ A B$ is a flat $B$-module by Algebra, Lemma 10.39.7. Hence we conclude by Lemma 15.66.3.
$\square$

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