Remark 24.35.5. Let $R$ be a ring and let $f_1, \ldots , f_ r \in R$ be a sequence of elements generating an ideal $I$. Let $K_ n$ be the Koszul complex on $f_1^ n, \ldots , f_ r^ n$ viewed as a differential graded $R$-algebra. We say $f_1, \ldots , f_ r$ is a *weakly proregular sequence* if for all $n$ there is an $m > n$ such that $K_ m \to K_ n$ induces the zero map on cohomology except in degree $0$. If so, then the arguments in the proof of Proposition 24.35.4 continue to work even when $R$ is not Noetherian. In particular we see that $\mathit{QC}(\{ R/I^ n\} )$ is equivalent as an $R$-linear triangulated category to the category $D_{comp}(R, I)$ of derived complete objects, provided $I$ can be generated by a weakly proregular sequence. If the need arises, we will precisely state and prove this here.

## 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Your email address will not be published.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)