The Stacks project

Lemma 15.66.10. Let $A \to B$ be a ring map. Let $K^\bullet $ and $L^\bullet $ be complexes of $B$-modules. Let $a, b, c, d \in \mathbf{Z}$. If

  1. $K^\bullet $ as a complex of $B$-modules has tor amplitude in $[a, b]$,

  2. $L^\bullet $ as a complex of $A$-modules has tor amplitude in $[c, d]$,

then $K^\bullet \otimes ^\mathbf {L}_ B L^\bullet $ as a complex of $A$-modules has tor amplitude in $[a + c, b + d]$.

Proof. We may assume that $K^\bullet $ is a complex of flat $B$-modules with $K^ i = 0$ for $i \not\in [a, b]$, see Lemma 15.66.3. Let $M$ be an $A$-module. Choose a free resolution $F^\bullet \to M$. Then

\[ (K^\bullet \otimes _ B^\mathbf {L} L^\bullet ) \otimes _ A^{\mathbf{L}} M = \text{Tot}(\text{Tot}(K^\bullet \otimes _ B L^\bullet ) \otimes _ A F^\bullet ) = \text{Tot}(K^\bullet \otimes _ B \text{Tot}(L^\bullet \otimes _ A F^\bullet )) \]

see Homology, Remark 12.18.4 for the second equality. By assumption (2) the complex $\text{Tot}(L^\bullet \otimes _ A F^\bullet )$ has nonzero cohomology only in degrees $[c, d]$. Hence the spectral sequence of Homology, Lemma 12.25.1 for the double complex $K^\bullet \otimes _ B \text{Tot}(L^\bullet \otimes _ A F^\bullet )$ proves that $(K^\bullet \otimes _ B^\mathbf {L} L^\bullet ) \otimes _ A^{\mathbf{L}} M$ has nonzero cohomology only in degrees $[a + c, b + d]$. $\square$


Comments (2)

Comment #5720 by on

Dear Joel, I checked the argument carefully and I do not think one needs this assumption. Also, in applications it is important that we don't assume this.

There are also:

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

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B66. Beware of the difference between the letter 'O' and the digit '0'.