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AC-GORENSTEIN RINGS AND THEIR STABLE MODULE CATEGORIES

Part of: Homological methods Applied homological algebra and category theory Homological algebra

Published online by Cambridge University Press:  29 October 2018

JAMES GILLESPIE
JAMES GILLESPIE*
Affiliation:
Ramapo College of New Jersey, School of Theoretical and Applied Science, 505 Ramapo Valley Road, Mahwah, NJ 07430, USA email jgillesp@ramapo.edu
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Abstract

We introduce what is meant by an AC-Gorenstein ring. It is a generalized notion of Gorenstein ring that is compatible with the Gorenstein AC-injective and Gorenstein AC-projective modules of Bravo–Gillespie–Hovey. It is also compatible with the notion of $n$-coherent rings introduced by Bravo–Perez. So a $0$-coherent AC-Gorenstein ring is precisely a usual Gorenstein ring in the sense of Iwanaga, while a $1$-coherent AC-Gorenstein ring is precisely a Ding–Chen ring. We show that any AC-Gorenstein ring admits a stable module category that is compactly generated and is the homotopy category of two Quillen equivalent abelian model category structures. One is projective with cofibrant objects that are Gorenstein AC-projective modules while the other is an injective model structure with fibrant objects that are Gorenstein AC-injectives.

Keywords

Gorenstein ringabelian model categorystable module category

MSC classification

Primary: 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
Secondary: 18G25: Relative homological algebra, projective classes 55U35: Abstract and axiomatic homotopy theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 107 , Issue 2 , October 2019 , pp. 181 - 198
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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