Denoetherianizing Cohen–Macaulay rings

László Fuchs and Bruce Olberding

Vol. 303 (2019), No. 1, 133–164

DOI: 10.2140/pjm.2019.303.133

Abstract

We introduce a new class of commutative nonnoetherian rings, called $n$-subperfect rings, generalizing the almost perfect rings that have been studied recently by Fuchs and Salce. For an integer $n\ge 0$, the ring $R$ is said to be $n$-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 $ht\left(M\right)$-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.

Keywords

Perfect, subperfect, $n$-subperfect rings, regular sequence, unmixed, Cohen–Macaulay rings

Mathematical Subject Classification 2010

Primary: 13F99, 13H10

Secondary: 13C13

Milestones

Received: 6 January 2018

Revised: 18 April 2019

Accepted: 28 April 2019

Published: 21 December 2019