Lemma 15.31.1. Let $R$ be a ring. Let $f_1, \ldots , f_ r \in R$ be an Koszul-regular sequence. Then the extended alternating Čech complex $R \to \bigoplus \nolimits _{i_0} R_{f_{i_0}} \to \bigoplus \nolimits _{i_0 < i_1} R_{f_{i_0}f_{i_1}} \to \ldots \to R_{f_1\ldots f_ r}$ from Section 15.29 only has cohomology in degree $r$.

**Proof.**
By Lemma 15.30.4 and induction the sequence $f_1, \ldots , f_{r - 1}, f_ r^ n$ is Koszul regular for all $n \geq 1$. By Lemma 15.28.4 any permutation of a Koszul regular sequence is a Koszul regular sequence. Hence we see that we may replace any (or all) $f_ i$ by its $n$th power and still have a Koszul regular sequence. Thus $K_\bullet (R, f_1^ n, \ldots , f_ r^ n)$ has nonzero cohomology only in homological degree $0$. This implies what we want by Lemma 15.29.6.
$\square$

## 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Your email address will not be published.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)