Lemma 15.65.8. Let $R$ be a ring. Let $K^\bullet $ be a bounded complex of $R$-modules such that $K^ i$ has tor amplitude in $[a - i, b - i]$ for all $i$. Then $K^\bullet $ has tor amplitude in $[a, b]$. In particular if $K^\bullet $ is a finite complex of $R$-modules of finite tor dimension, then $K^\bullet $ has finite tor dimension.

**Proof.**
Follows by induction on the length of the finite complex: use Lemma 15.65.5 and the stupid truncations.
$\square$

## 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Your email address will not be published.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.

## Comments (0)

There are also: