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57
Stable model categories are categories of modules
 TOPOLOGY
, 2003
"... A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for ..."
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Cited by 78 (16 self)
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A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the EilenbergMac Lane spectrum HR and (unbounded) chain complexes of Rmodules for a ring R.
Noncommutative curves and noncommutative surfaces
 Bulletin of the American Mathematical Society
"... Abstract. In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative grad ..."
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Cited by 48 (6 self)
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Abstract. In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing
Cluster Categories for Algebras of Global Dimension 2 and . . .
, 2008
"... Let k be a field and A a finitedimensional kalgebra of global dimension ≤ 2. We construct a triangulated category CA associated to A which, if A is hereditary, is triangle equivalent to the cluster category of A. When CA is Homfinite, we prove that it is 2CY and endowed with a canonical cluster ..."
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Cited by 33 (1 self)
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Let k be a field and A a finitedimensional kalgebra of global dimension ≤ 2. We construct a triangulated category CA associated to A which, if A is hereditary, is triangle equivalent to the cluster category of A. When CA is Homfinite, we prove that it is 2CY and endowed with a canonical clustertilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by GeissLeclercSchröer and by BuanIyamaReitenScott. Our results rely on quivers with potential. Namely, we introduce a cluster category C (Q,W) associated to a quiver with potential (Q, W). When it is Jacobifinite we prove that it is endowed with a clustertilting object whose endomorphism algebra is isomorphic
Ktheory and derived equivalences
 Duke Math. J
"... Abstract. We show that if two rings have equivalent derived categories then they have the same algebraic Ktheory. Similar results are given for Gtheory, and for a large class of abelian categories. Contents ..."
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Cited by 29 (6 self)
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Abstract. We show that if two rings have equivalent derived categories then they have the same algebraic Ktheory. Similar results are given for Gtheory, and for a large class of abelian categories. Contents
Realizability Of Modules Over Tate Cohomology
, 2001
"... Let k be a eld and let G be a nite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology G 2 HH 3; 1 ^ H (G; k) with the following property. Given a graded ^ H (G; k)module X, the image of G in Ext 3; 1 ^ H (G;k) (X; X) vanishes if and only if X is is ..."
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Cited by 20 (1 self)
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Let k be a eld and let G be a nite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology G 2 HH 3; 1 ^ H (G; k) with the following property. Given a graded ^ H (G; k)module X, the image of G in Ext 3; 1 ^ H (G;k) (X; X) vanishes if and only if X is isomorphic to a direct summand of ^ H (G; M) for some kGmodule M . The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a dierential graded algebra A, there is also a canonical element of Hochschild cohomology HH 3; 1 H (A) which is a predecessor for these obstructions.
Morita theory in abelian, derived and stable model categories, Structured ring spectra
 London Math. Soc. Lecture Note Ser
, 2004
"... These notes are based on lectures given at the Workshop on Structured ring spectra and ..."
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Cited by 19 (0 self)
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These notes are based on lectures given at the Workshop on Structured ring spectra and
Derived equivalences from mutations of quivers with potential
 ADVANCES IN MATHEMATICS 226 (2011) 2118–2168
, 2011
"... ..."