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. $\square$

There are also:

• 2 comment(s) on Section 15.66: Tor dimension

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).