The Stacks project

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.

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