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Finite Gorenstein representation type implies simple singularity
, 2008
"... Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive Rmodules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity ..."
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Cited by 17 (6 self)
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Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive Rmodules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to argue that if the residue field has a totally reflexive cover, then R is Gorenstein or every totally reflexive Rmodule is free.
Support varieties and representation type of small quantum groups
 Internat. Math. Res. Notices
"... Abstract. In this paper we provide a wildness criterion for any finite dimensional Hopf algebra with finitely generated cohomology. This generalizes a result of Farnsteiner to not necessarily cocommutative Hopf algebras over ground fields of arbitrary characteristic. Our proof uses the theory of su ..."
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Cited by 14 (6 self)
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Abstract. In this paper we provide a wildness criterion for any finite dimensional Hopf algebra with finitely generated cohomology. This generalizes a result of Farnsteiner to not necessarily cocommutative Hopf algebras over ground fields of arbitrary characteristic. Our proof uses the theory of support varieties for modules, one of the crucial ingredients being a tensor product property for some special modules. As an application we prove a conjecture of Cibils stating that small quantum groups of rank at least two are wild. 1.
Construction of totally reflexive modules from an exact pair of zero divisors
 Bull. London Math. Soc
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Homotopy coherent centers versus centers of homotopy categories
 IN 46556 USA DWYER.1@ND.EDU MARKUS SZYMIK DEPARTMENT OF MATHEMATICAL SCIENCES NTNU NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY 7491 TRONDHEIM NORWAY MARKUS.SZYMIK@MATH.NTNU.NO
, 2013
"... Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as Hochschild cohomology type invariants in contexts that are no ..."
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Cited by 2 (2 self)
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Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as Hochschild cohomology type invariants in contexts that are not necessarily linear or stable, and we argue that they are more appropriate to higher categorical contexts than the centers of their homotopy or derived categories. Among many other things, we present an obstruction theory for realizing elements in the centers of homotopy categories, and a BousfieldKan type spectral sequence that computes the homotopy groups. Nontrivial classes of examples are given as illustration throughout.
Current Research: Noncommutative Representation Theory
"... Group actions are ubiquitous in mathematics. To understand a mathematical object, it is often helpful to understand its symmetries as expressed by a group. For example, a group acts on a ring by automorphisms (preserving its structure). Analogously, a Lie algebra acts on a ring by derivations. Unify ..."
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Group actions are ubiquitous in mathematics. To understand a mathematical object, it is often helpful to understand its symmetries as expressed by a group. For example, a group acts on a ring by automorphisms (preserving its structure). Analogously, a Lie algebra acts on a ring by derivations. Unifying these two types of actions are Hopf algebras acting on rings. A Hopf algebra is not only an algebra, but also a coalgebra, and the notion of an action preserving the structure of a ring uses this property. The