Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend

We investigate the (unbounded) derived category of a differential Z-graded 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 Z-graded k-algebra 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) A-module is a Z-graded A-module 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 A-modules is a homogeneous morphism of degree 0 of the underlying graded A-modules commuting with the differentials. The DG A-modules form an abelian category CA. A morphism f : M ! N of CA is null-homotopic if f = dr + rd for some homogeneous morphism r : M ! N of degree-1 of the underlying graded A-modules.

Abstract. We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay modules is 3-Calabi-Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [27]. In addition, we prove a general result about relative 3-Calabi-Yau duality over non stable endomorphism rings. This strengthens and generalizes the Ext-group symmetries obtained in [27] for simple modules. Finally, we generalize the results on relative Calabi-Yau duality from 2-Calabi-Yau to d-Calabi-Yau categories. We show how to produce many examples of d-cluster tilted algebras. 1.

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.

Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau 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

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 Thomason-Trobaugh’s [51] and Yao’s [58] localization theorems in K-theory (cf. also [55]).

These notes are based on a series of five lectures given during the

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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 p-adic integers Zp and finite p-groups. In the subsequent article [14] it is shown that if Λ is a representation-directed 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 Ringel-Hall 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 representation-finite hereditary k-algebra 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 Z-form of n+, and Z ⊗ Z[T]H(Λ) becomes the Kostant Z-form of the universal enveloping algebra U(n+) [15]. For a general finite dimensional hereditary k-algebra 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 Kac-Moody Lie algebra via the module categories of finite dimensional hereditary k-algebras. 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