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AC-GORENSTEIN RINGS AND THEIR STABLE MODULE CATEGORIES

Published online by Cambridge University Press:  29 October 2018

JAMES GILLESPIE*
Affiliation:
Ramapo College of New Jersey, School of Theoretical and Applied Science, 505 Ramapo Valley Road, Mahwah, NJ 07430, USA email [email protected]
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Abstract

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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.

Type
Research Article
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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