Lemma 15.66.11. Let $A \to B$ be a ring map. Assume that $B$ is flat as an $A$-module. Let $K^\bullet$ be a complex of $B$-modules. Let $a, b \in \mathbf{Z}$. If $K^\bullet$ as a complex of $B$-modules has tor amplitude in $[a, b]$, then $K^\bullet$ as a complex of $A$-modules has tor amplitude in $[a, b]$.

Proof. This is a special case of Lemma 15.66.10, but can also be seen directly as follows. We have $K^\bullet \otimes _ A^{\mathbf{L}} M = K^\bullet \otimes _ B^{\mathbf{L}} (M \otimes _ A B)$ since any projective resolution of $K^\bullet$ as a complex of $B$-modules is a flat resolution of $K^\bullet$ as a complex of $A$-modules and can be used to compute $K^\bullet \otimes _ A^{\mathbf{L}} M$. $\square$

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