Eduardo N. Marcos

Skew category, Galois covering and smash product of a k-category

In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In the case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beatties construction of a ring with local units. We recover in a categorical generalized setting the Duality Theorems of M. Cohen and S. Montgomery (1984), and we provide a unification with the results on coverings of quivers and relations by E. Green (1983). We obtain a confirmation in a quiver and relations-free categorical setting that both constructions are mutual inverses, namely the quotient of a free action category and the smash product of a graded category. Finally we describe functorial relations between the representation theories of a category and of a Galois cover of it.

δ-Koszul Algebras

ABSTRACT Let A = A 0 ⊕ A 1 ⊕ A 2 ⊕ ··· be a graded K -algebra such that A 0 is a finite product of copies of the field K, A is generated in degrees 0 and 1,and dim K A 1 < ∞. We study those graded algebras A with the property that A 0 , viewed as a graded A -module, has a graded projective resolution, , such that each P i can be generated in a single degree. The paper describes necessary and sufficient conditions for the Ext-algebra of A , , to be finitely generated. We also investigate classes of modules over such algebras and Veronese subrings of the Ext-algebra.

On the hochschild cohomology of truncated cycle algebras

The purpose of this paper is to study the Hochschild cohomology ring H *(⋀)of algebras of the form ⋀ = kZe /JN , where Ze is an oriented cycle with e vertices and J is the ideal generated by the arrows, N≥2. We provide a new description of the Yoneda product in H *(⋀). and prove that this is a finitely generated infinite dimensional ring. In addition we show that algebras of the form ⋀ = kZe /JN are not derived equivalent unless they are isomorphic.

Resolutions over Koszul algebras

Abstract.In this paper we show that if

The Hochschild Cohomology Ring of a One Point Extension

\Lambda = \mathop \coprod \limits_{i \geqq 0} \Lambda _i

Hochschild Cohomology of skew group rings and invariants

is a Koszul algebra with Λ0 isomorphic to a product of copies of a field, then the minimal projective resolution of Λ0 as a right Λ-module provides all the information necessary to construct both a minimal projective resolution of Λ0 as a left Λ-module and a minimal projective resolution of Λ as a right module over the enveloping algebra of Λ. The main tool for this is showing that there is a comultiplicative structure on a minimal projective resolution of Λ0 as a right Λ-module.

Cohomology of split algebras and of trivial extensions

It is well known that an artin algebra A is of finite representation type if and only if red(modA)=0. In this note we deepen this result by showing that(rad(modA))2=0 implies that A is of finite representation type.

Diagonalizable derivations of finite-dimensional algebras I

Abstract This paper studies the ring structure of the Hochschild cohomology ring of an algebra. The first main result gives a ring homomorphism from the Hochschild cohomology ring of an algebra A to the Ext-algebra of an A-module. Then, for a one point extension B of a finite dimensional algebra A, we relate the ring structures of the Hochschild cohomology rings of A and B thus giving a method of constructing algebras with non-trivial ring structure on the Hochschild cohomology ring.