Mathematics – Algebraic Geometry
Scientific paper
2010-07-08
Mathematics
Algebraic Geometry
33 pages, typos fixed, some minor changes
Scientific paper
We first generalize classical Auslander--Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of SpecR, in the main body of this paper we introduce a theory of modifying and maximal modifying modules for three-dimensional Gorenstein rings R. Under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules are shown to be derived equivalent. We then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory. The behavior of our mutation strongly depends on whether a certain factor algebra is artinian --- when it is not artinian our mutation may be the identity.
Iyama Osamu
Wemyss Michael
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