Roberto Martínez-Villa

Approximations with modules having linear resolutions

Abstract Let Λ be a Koszul algebra over a field K . We study in this paper a class of modules closely related to the Koszul modules called weakly Koszul modules. It turns out that these modules have some special filtrations with modules having linear resolutions and therefore easy to describe minimal projective resolutions. We prove that if the Koszul dual of a finite-dimensional Koszul algebra is Noetherian then every finitely generated graded module has a weakly Koszul syzygy and as a consequence a rational Poincare series. If Λ is selfinjective Koszul, we prove that the stable part of each connected component of the graded Auslander–Reiten quiver containing a weakly Koszul module is of the form Z A ∞ , and if the Koszul dual of Λ is Noetherian, then every component has its stable part of the form Z A ∞ .

Hochschild Cohomology of skew group rings and invariants

Let A be a k-algebra and G be a group acting on A. We show that G also acts on the Hochschild cohomology algebra HH⊙(A) and that there is a monomorphism of rings HH⊙(A)G→HH⊙(A[G]). That allows us to show the existence of a monomorphism from HH⊙(Ã)G into HH⊙(A), where à is a Galois covering with group G.

Graded and Koszul Categories

Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there is always a naturally associated Koszul theory. To obtain this, the notions of Koszul algebras, linear modules and Koszul duality are extended to additive (graded) categories over a field. The main focus of this paper is to provide these generalizations and the necessary preliminaries.

Auslander–Reiten sequences, locally free sheaves and Chebysheff polynomials

Let R be the exterior algebra in n + 1 variables, and let S denote the symmetric algebra in n + 1 variables. It is well known that R is a selfinjective Koszul algebra and S is its Koszul dual. By KR and KS we denote the categories of linear R-modules (S-modules respectively) where the morphisms are the degree zero homomorphisms. The Koszul duality can be then used to obtain mutually inverse dualities between the category of linear R-modules and that of the linear S-modules:

Tilting Theory and Functor Categories I. Classical Tilting

Tilting theory has been a very important tool in the classification of finite dimensional algebras of finite and tame representation type, as well as, in many other branches of mathematics. Happel (1988) and Cline et al. (J Algebra 304:397–409 1986) proved that generalized tilting induces derived equivalences between module categories, and tilting complexes were used by Rickard (J Lond Math Soc 39:436–456, 1989) to develop a general Morita theory of derived categories. On the other hand, functor categories were introduced in representation theory by Auslander (I Commun Algebra 1(3):177–268, 1974), Auslander (1971) and used in his proof of the first Brauer–Thrall conjecture (Auslander 1978) and later on, used systematically in his joint work with I. Reiten on stable equivalence (Auslander and Reiten, Adv Math 12(3):306–366, 1974), Auslander and Reiten (1973) and many other applications. Recently, functor categories were used in Martínez-Villa and Solberg (J Algebra 323(5):1369–1407, 2010) to study the Auslander–Reiten components of finite dimensional algebras. The aim of this paper is to extend tilting theory to arbitrary functor categories, having in mind applications to the functor category Mod (modΛ), with Λ a finite dimensional algebra.

Tilting Theory and Functor Categories II. Generalized Tilting

In this paper we continue the project of generalizing tilting theory to the category of contravariant functors

Contravariantly Finite Subcategories and Torsion Theories

\mathrm{Mod}(\mathcal{C})

D -Koszul algebras

, from a skeletally small preadditive category

On Modules with Linear Presentations

\mathcal{C}

Graded, Selfinjective, and Koszul Algebras

to the category of abelian groups, initiated in [15]. We introduce the notion of a generalized tilting category