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Frobenius categories, Gorenstein algebras and rational surface singularities

Part of: Surfaces and higher-dimensional varieties Theory of modules and ideals Abelian categories Homological methods

Published online by Cambridge University Press:  28 October 2014

Martin Kalck ,
Osamu Iyama ,
Michael Wemyss  and
Dong Yang
Martin Kalck
Affiliation:
The Maxwell Institute, School of Mathematics, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK email m.kalck@ed.ac.uk
Osamu Iyama
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan email iyama@math.nagoya-u.ac.jp
Michael Wemyss
Affiliation:
The Maxwell Institute, School of Mathematics, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK email wemyss.m@googlemail.com
Dong Yang
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, PR China email dongyang2002@gmail.com
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Abstract

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We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga–Gorenstein ring. We then apply this result to the Frobenius category of special Cohen–Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga–Gorenstein rings of finite Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander–Solberg and Kong type results.

Keywords

Frobenius categoriesIwanaga–Gorenstein algebrasGorenstein-projective modulesCohen–Macaulay modulesnon-commutative resolutionssingularity categoriesrational surface singularities

MSC classification

Primary: 14J17: Singularities 13C14: Cohen-Macaulay modules 18E30: Derived categories, triangulated categories 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 3 , March 2015 , pp. 502 - 534
Copyright
© The Author(s) 2014 

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