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102
Triangulated categories of singularities and Dbranes in LandauGinzburg models
, 2003
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Deriving Dg Categories
, 1993
"... We investigate the (unbounded) derived category of a differential Zgraded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categori ..."
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Cited by 123 (9 self)
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We investigate the (unbounded) derived category of a differential Zgraded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a 'Morita theorem` (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories. Summary We give an account of the contents of this paper for the special case of DG algebras. Let k be a commutative ring and A a DG (k)algebra, i.e. a Zgraded kalgebra A = a p2Z A p endowed with a differential d of degree 1 such that d(ab) = (da)b + (\Gamma1) p a(db) for all a 2 A p , b 2 A. A DG (right) Amodule is a Zgraded Amodule M = ` p2Z M p endowed with a differential d of degree 1 such that d(ma) = (dm)a + (\Gamma1) p m(da) for all m 2 M p , a 2 A. A morphism of DG Amodules is a homogeneous morphism of degree 0 of the underlying graded Amodules commuting with the differentials. The DG Amodules form an abelian category CA. A morphism f : M ! N of CA is nullhomotopic if f = dr + rd for some homogeneous morphism r : M ! N of degree1 of the underlying graded Amodules.
Idempotent completion of triangulated categories
 J. Algebra
, 2001
"... Abstract. We show that the idempotent completion of a triangulated category has a natural structure of ..."
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Cited by 109 (11 self)
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Abstract. We show that the idempotent completion of a triangulated category has a natural structure of
Cluster structures for 2CalabiYau categories and unipotent groups
"... Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This c ..."
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Cited by 108 (19 self)
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Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This class of 2CalabiYau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2CalabiYau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related
On the Cyclic Homology of Exact Categories
 JPAA
"... The cyclic homology of an exact category was defined by R. McCarthy [26] using the methods of F. Waldhausen [36]. McCarthy's theory enjoys a number of desirable properties, the most basic being the agreement property, i.e. the fact that when applied to the category of finitely generated project ..."
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Cited by 88 (1 self)
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The cyclic homology of an exact category was defined by R. McCarthy [26] using the methods of F. Waldhausen [36]. McCarthy's theory enjoys a number of desirable properties, the most basic being the agreement property, i.e. the fact that when applied to the category of finitely generated projective modules over an algebra it specializes to the cyclic homology of the algebra. However, we show that McCarthy's theory cannot be both compatible with localizations and invariant under functors inducing equivalences in the derived category. This is our motivation for introducing a new theory for which all three properties hold: extension, invariance and localization. Thanks to these properties, the new theory can be computed explicitly for a number of categories of modules and sheaves.
Compact generators in categories of matrix factorizations
 MR2824483 (2012h:18014), Zbl 1252.18026, arXiv:0904.4713
"... Abstract. We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. We exhibit the stabilized residue field as a compact generator. This implies a quasiequivalence between the category of matrix factorizations and the dg derived category of an ex ..."
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Cited by 52 (1 self)
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Abstract. We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. We exhibit the stabilized residue field as a compact generator. This implies a quasiequivalence between the category of matrix factorizations and the dg derived category of an explicitly computable dg algebra. Building on this quasiequivalence we establish a derived Morita theory which identifies the functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain complexes of matrix factorization categories. Finally, we give interpretations of the results of this work in terms of noncommutative geometry modelled on dg categories. Contents
Invariance and localization for cyclic homology of DG algebras
 J. PURE APPL. ALGEBRA
, 1998
"... We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary ’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyc ..."
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Cited by 49 (6 self)
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We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary ’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyclic homology. This completes results of Rickard’s [48] and Happel’s [18]. It also extends well known results on preservation of cyclic homology under Morita equivalence [10], [39], [25], [26], [41], [42]. We then show that under suitable flatness hypotheses, an exact sequence of derived categories of DG algebras yields a long exact sequence in cyclic homology. This may be viewed as an analogue of ThomasonTrobaugh’s [51] and Yao’s [58] localization theorems in Ktheory (cf. also [55]).
Derived categories, resolutions, and Brown representability. In: Interactions between homotopy theory and algebra, 101–139,
 Contemp. Math. 436, Amer. Math. Soc.,
, 2007
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From triangulated categories to Lie algebras: A theorem of Peng and Xiao
, 2006
"... In his seminal article [13], Ringel showed how to associate to any finitary ring Λ an associative unital algebra H(Λ), with structure constants encoding information about extensions between finite modules. This generalised the Hall algebra [3, 17], which deals with the ring of padic integers Zp and ..."
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Cited by 25 (1 self)
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In his seminal article [13], Ringel showed how to associate to any finitary ring Λ an associative unital algebra H(Λ), with structure constants encoding information about extensions between finite modules. This generalised the Hall algebra [3, 17], which deals with the ring of padic integers Zp and finite pgroups. In the subsequent article [14] it is shown that if Λ is a representationdirected algebra over a finite field k, then the structure constants are given by evaluating integer polynomials. Using these Hall polynomials as structure constants, one may therefore form the generic RingelHall algebra over Z[T]. Let n(Λ) be the subgroup of H(Λ) generated by the indecomposable modules. If we specialise T ↦ → 1, then Z ⊗ Z[T]n(Λ) becomes a Lie subalgebra of Z ⊗ Z[T]H(Λ). In fact, over the rational numbers, Q ⊗ Z[T] H(Λ) is isomorphic to the universal enveloping algebra of Q ⊗ Z[T] n(Λ). In particular, let Λ be a representationfinite hereditary kalgebra and let g = n − ⊕h⊕n+ be the semisimple complex Lie algebra of the same type as Λ. Then Z ⊗ Z[T]n(Λ) can be identified with the Chevalley Zform of n+, and Z ⊗ Z[T]H(Λ) becomes the Kostant Zform of the universal enveloping algebra U(n+) [15]. For a general finite dimensional hereditary kalgebra Λ one considers the composition algebra, the subalgebra generated by the simple modules. This also has a generic version, constructed as a subalgebra of a direct product over infinitely many finite fields of composition algebras [16]. Green showed in [2] that the generic composition algebra (after twisting the multiplication via the Euler form of the category modΛ) is isomorphic to the quantum group of the same type as Λ. Therefore, we can realise the quantum group of any symmetrisable KacMoody Lie algebra via the module categories of finite dimensional hereditary kalgebras. A natural question is whether it is possible to obtain the full (quantised) enveloping algebra, or at least the full Lie algebra. The latter question was answered by Peng and Xiao in [9] for the affine Lie algebras of type Ã, and in [10] for the