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Beyond totally reflexive modules and back  A survey on Gorenstein dimensions
, 2009
"... Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory’s connections with relative homological algebra and with studies of local ring homomorphisms. It ends close to the star ..."
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Cited by 10 (0 self)
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Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory’s connections with relative homological algebra and with studies of local ring homomorphisms. It ends close to the starting point: with a characterization of Gorenstein rings in terms of total acyclicity of complexes.
Rings without a Gorenstein analogue of the GovorovLazard theorem
, 2008
"... It was proved by Beligiannis and Krause that over certain Artin algebras, there are Gorenstein flat modules which are not direct limits of finitely generated Gorenstein projective modules. That is, these algebras have no Gorenstein analogue of the GovorovLazard Theorem. We show that, in fact, the ..."
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Cited by 4 (2 self)
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It was proved by Beligiannis and Krause that over certain Artin algebras, there are Gorenstein flat modules which are not direct limits of finitely generated Gorenstein projective modules. That is, these algebras have no Gorenstein analogue of the GovorovLazard Theorem. We show that, in fact, there is a large class of rings without such an analogue. Namely, let R be a commutative local noetherian ring. Then the analogue fails for R if it has a dualizing complex, is henselian, not Gorenstein, and has a finitely generated Gorenstein projective module which is not free. The proof is based on a theory of Gorenstein projective (pre)envelopes. We show, among other things, that the finitely generated Gorenstein projective modules form an enveloping class in mod R if and only if R is Gorenstein or has the property that each finitely generated Gorenstein projective module is free.
Gorenstein homology, relative pure homology and virtually Gorenstein rings
 J PURE APPL. ALGEBRA
, 2014
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