Lemma 15.66.12. Let $A \to B$ be a ring map. Assume that $B$ has tor dimension $\leq d$ 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 $[a - d, b]$.
Proof. This is a special case of Lemma 15.66.10, but can also be seen directly as follows. Let $M$ be an $A$-module. Choose a free resolution $F^\bullet \to M$. Then
\[ K^\bullet \otimes _ A^{\mathbf{L}} M = \text{Tot}(K^\bullet \otimes _ A F^\bullet ) = \text{Tot}(K^\bullet \otimes _ B (F^\bullet \otimes _ A B)) = K^\bullet \otimes _ B^{\mathbf{L}} (M \otimes _ A^{\mathbf{L}} B). \]
By our assumption on $B$ as an $A$-module we see that $M \otimes _ A^{\mathbf{L}} B$ has cohomology only in degrees $-d, -d + 1, \ldots , 0$. Because $K^\bullet $ has tor amplitude in $[a, b]$ we see from the spectral sequence in Example 15.62.4 that $K^\bullet \otimes _ B^{\mathbf{L}} (M \otimes _ A^{\mathbf{L}} B)$ has cohomology only in degrees $[-d + a, b]$ as desired. $\square$
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