Positive Galois coverings of self-injective algebras

In the article, we study the structure of Galois coverings of self-injective artin algebras with infinite cyclic Galois groups. In particular, we characterize all basic, connected, self-injective artin algebras having Galois coverings by the repetitive algebras of basic connected artin algebras and with the Galois groups generated by positive automorphisms of the repetitive algebras.     

Koszul algebras and finite Galois coverings

The relationships between Koszulity and finite Galois coverings are obtained, which provide a construction of Koszul algebras by finite Galois coverings.     

Hochschild （Co）homology of Galois Coverings of Grassmann Algebras

Let ∧ be the Z2-Galois covering of the Grassmann algebra A over a field k of characteristic not equal to 2. In this paper, the dimensional formulae of Hochschild homology and cohomology groups of ∧ are calculated explicitly. And the cyclic homology of∧ can also be calculated when the underlying field is of characteristic zero. As a result, we prove that there is an isomorphism from i≥1 HH^i（∧） to i≥1 HH^i（∧）.     

Derived equivalences of selfinjective algebras preserve singularities

We prove that the derived equivalences (more generally the stable equivalences of Morita type) of finite dimensional selfinjective algebras over algebraically closed fields preserve the types of singularities in the orbit closures of module varieties. As an application, we obtain that the orbit closures in the module varieties of the Brauer tree algebras are normal and Cohen-Macaulay. Mathematics Subject Classification (2000):14B05, 14L30, 16D50, 16G20     

Quantitative Siegel's theorem for Galois coverings

It is known that Siegels theorem on integral points is effective for Galoiscoverings of the projective line. In this paper we obtain a quantitative version of this result, giving an explicit upper bound for the heights of S-integral K-rational points in terms of the number field K, the set of places S and the defining equation of the curve.Our main tools are Bakers theory of linear forms in logarithms and thequantitative Eisenstein theorem due to Schmidt, Dwork and van der Poorten.     

Cartan matrices of selfinjective algebras of tubular type

The Cartan matrix of a finite dimensional algebra A is an important combinatorial invariant reflecting frequently structural properties of the algebra and its module category. For example, one of the important features of the modular representation theory of finite groups is the nonsingularity of Cartan matrices of the associated group algebras (Brauer’s theorem). Recently, the class of all tame selfinjective algebras having simply connected Galois coverings and the stable Auslander-Reiten quiver consisting only of stable tubes has been shown to be the class of selfinjective algebras of tubular type, that is, the orbit algebras /G of the repetitive algebras of tubular algebras B with respect to the actions of admissible groups G of automorphisms of . The aim of the paper is to describe the determinants of the Cartan matrices of selfinjective algebras of tubular type and derive some consequences.     

Radical cube zero selfinjective algebras of finite complexity

One of our main results is a classification of all the possible quivers of selfinjective radical cube zero finite-dimensional algebras over an algebraically closed field having finite complexity. In the paper (Erdmann and Solberg, 2011) [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite-dimensional algebra to have such a theory of support varieties implies that the algebra has finite complexity. Hence this paper is a partial extension of [5].     

Galois coverings and endomorphisms of projective varieties

We prove that the vector bundle associated to a Galois covering of projective manifolds is ample (resp. nef) under very mild conditions. This results is applied to the study of ramified endomorphisms of Fano manifolds with b ₂ = 1. It is conjectured that is the only Fano manifold admitting an endomorphism of degree d ≥ 2, and we verify this conjecture in several cases. An important ingredient is a generalization of a theorem of Andreatta–Wisniewski, characterizing projective space via the existence of an ample subsheaf in the tangent bundle. Marian Aprodu was supported in part by a Humboldt Research Fellowship and a Humboldt Return Fellowship. He expresses his special thanks to the Mathematical Institute of Bayreuth University for hospitality during the first stage of this work. Stefan Kebekus and Thomas Peternell were supported by the DFG-Schwerpunkt “Globale Methoden in der komplexen Geometrie” and the DFG-Forschergruppe “Classification of Algebraic Surfaces and Compact Complex Manifolds”. A part of this paper was worked out while Stefan Kebekus visited the Korea Institute for Advanced Study. He would like to thank Jun-Muk Hwang for the invitation.     

On the Galois coverings of a cluster-tilted algebra

We study the module category of a certain Galois covering of a cluster-tilted algebra which we call the cluster repetitive algebra. Our main result compares the module categories of the cluster repetitive algebra of a tilted algebra C and the repetitive algebra of C, in the sense of Hughes and Waschbüsch.     

Torsion theories and Galois coverings of topological groups

For any torsion theory in a homological category, one can define a categorical Galois structure and try to describe the corresponding Galois coverings. In this article we provide several characterizations of these coverings for a special class of torsion theories, which we call quasi-hereditary. We describe a new reflective factorization system that is induced by any quasi-hereditary torsion theory. These results are then applied to study various examples of torsion theories in the category of topological groups.     

A characterization of selfinjective algebras of strictly canonical type

We prove that the class of selfinjective algebras of strictly canonical type, investigated in Kwiecień and Skowroński (2009) [27], Kwiecień and Skowroński (2009) [28], coincides with the class of selfinjective algebras having triangular Galois coverings with infinite cyclic group and the Auslander–Reiten quiver with quasi-tubes maximally saturated by simple and projective modules, satisfying natural conditions.     

Galois coverings of the product of the projective lines by abelian surfaces

We consider the quotient space of an abelian surface by a finite subgroup of the automorphism group. We classify the analytic representation of the group and the branch divisor of the natural projection to the quotient space, where the quotient space is isomorphic to the product of the projective lines.