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arXiv · 1712.01753

De-noetherizing Cohen-Macaulay rings

Laszlo Fuchs·Bruce Olberding
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Abstract

We introduce a new class of commutative {non-noetherian} rings, called $n$-subperfect rings, generalizing the almost perfect rings that have been studied recently by Fuchs-Salce. For an integer $n \ge 0$, the ring $R$ is $n$-subperfect if every maximal regular sequence in $R$ has length $n$ and the total ring of quotients of $R/I$ for any ideal $I$ generated by a regular sequence is a perfect ring in the sense of Bass. We define an extended Cohen-Macaulay ring as a commutative ring $R$ that has noetherian prime spectrum and each localization $R_M$ at a maximal ideal $M$ is ht($M$)-subperfect. In the noetherian case, these are precisely the classical Cohen-Macaulay rings. Several relevant properties are proved reminiscent of those shared by Cohen-Macaulay rings.

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Laszlo Fuchs, Bruce Olberding. 2017-12-05. De-noetherizing Cohen-Macaulay rings. https://arxiv.org/abs/1712.01753

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