Derived equivalences and Gorenstein algebras

We introduce a notion of Gorenstein R-algebras over a commutative Gorenstein ring R. Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein R-algebra A to have a Gorenstein R-algebra B as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein R-algebra A whose endomorphism algebra is not a Gorenstein R-algebra.     

The quiver of projectives in hereditary categories with Serre duality

Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to for a so-called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form ZQ.     

Derived equivalences between associative deformations

We prove that if two associative deformations (parameterized by the same complete local ring) are derived Morita equivalent, then they are Morita equivalent (in the classical sense).     

From iterated tilted algebras to cluster-tilted algebras

In this paper the relationship between iterated tilted algebras and cluster-tilted algebras and relation extensions is studied. In the Dynkin case, it is shown that the relationship is very strong and combinatorial.     

Cluster-tilted algebras are Gorenstein and stably Calabi-Yau

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 [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press]. 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 [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press] 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.     

Stable functors of derived equivalences and Gorenstein projective modules

From certain triangle functors, called nonnegative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of nonnegative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. In particular, we prove that the stable functor of a derived equivalence between two arbitrary rings provides an explicit triangle equivalence between the stable categories of Gorenstein projective modules. This generalizes a result of Y. Kato. Our results can also be applied to provide shorter proofs of some known results on homological conjectures.     

Constructions of stable equivalences of Morita type for finite dimensional algebras II

Suppose k is a field. Let A and B be two finite dimensional k-algebras such that there is a stable equivalence of Morita type between A and B. In this paper, we prove that (1) if A and B are representation-finite then their Auslander algebras are stably equivalent of Morita type; (2) The n-th Hochschild homology groups of A and B are isomorphic for all n≥1. A new proof is also provided for Hochschild cohomology groups of self-injective algebras under a stable equivalence of Morita type.     

Vanishing of self-extensions over symmetric algebras

We study self-extensions of modules over symmetric artin algebras. We show that non-projective modules with eventually vanishing self-extensions must lie in AR components of stable type Z_A∞$\mathbb{Z}{A}_{\infty }$. Moreover, the degree of the highest non-vanishing self-extension of these modules is determined by their quasilength. This has implications for the Auslander–Reiten Conjecture and the Extension Conjecture.     

Grothendieck group and generalized mutation rule for 2-Calabi-Yau triangulated categories

We compute the Grothendieck group of certain 2-Calabi-Yau triangulated categories appearing naturally in the study of the link between quiver representations and Fomin-Zelevinsky cluster algebras. In this setup, we also prove a generalization of the Fomin-Zelevinsky mutation rule.     

Universal constructions for Hopf algebras

The category of Hopf monoids over an arbitrary symmetric monoidal category as well as its subcategories of commutative and cocommutative objects respectively are studied, where attention is paid in particular to the following questions: (a) When are the canonical forgetful functors of these categories into the categories of monoids and comonoids respectively part of an adjunction? (b) When are the various subcategory-embeddings arsing naturally in this context reflexive or coreflexive? (c) When does a category of Hopf monoids have all limits or colimits? These problems are also shown to be intimately related. Particular emphasis is given to the case of Hopf algebras, i.e., when the chosen symmetric monoidal category is the category of modules over a commutative unital ring.     

Combinatorial derived invariants for gentle algebras

We define derived equivalent invariants for gentle algebras, constructed in an easy combinatorial way from the quiver with relations defining these algebras. Our invariants consist of pairs of natural numbers and contain important information about the algebra and the structure of the stable Auslander-Reiten quiver of its repetitive algebra. As a by-product we obtain that the number of arrows of the quiver of a gentle algebra is invariant under derived equivalence. Finally, our invariants separate the derived equivalence classes of gentle algebras with at most one cycle.     

Artin-Schelter regular algebras and categories

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin-Schelter regular algebras of dimension n to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin-Schelter regular algebra or a category of dimension n is shown to have common properties with the classical Artin-Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the -category of nice sets of simple objects of maximal projective dimension n is a finite length Frobenius category.     

Derived categories of graded gentle one-cycle algebras

Let A be a graded algebra. It is shown that the derived category of dg modules over A (viewed as a dg algebra with trivial differential) is a triangulated hull of a certain orbit category of the derived category of graded A-modules. This is applied to study derived categories of graded gentle one-cycle algebras.     

Graded Calabi Yau algebras of dimension 3

In this paper, we prove that Graded Calabi Yau algebras of dimension 3 are isomorphic to path algebras of quivers with relations derived from a superpotential. We show that for a given quiver Q and a degree d, the set of good superpotentials of degree d, i.e. those that give rise to Calabi Yau algebras, is either empty or almost everything (in the measure theoretic sense). We also give some constraints on the structure of quivers that allow good superpotentials, and for the simplest quivers we give a complete list of the degrees for which good superpotentials exist.