María Inés Platzeck

MODULES OF FINITE PROJECTIVE DIMENSION FOR STANDARDLY STRATIFIED ALGEBRAS

This paper deals with standardly stratified algebras, a generalization of quasihereditary algebras. As for quasihereditary algebras we show that there is a tilting module naturally associated with the category F(Δ) of modules with Δ-filtration. We find sufficient conditions in terms of quivers with relations for F(Δ) to coincide with the category P <∞(Λ) of modules of finite projective dimension, thus generalizing earlier results and providing new classes of examples where P <∞(Λ) is contravariantly finite. We also prove that for a standardly stratified algebra the relations for the Grothendieck group for F(Δ) are generated by almost split sequences.

On a theorem of auslander and applications

Throughout t,his paper A will denote an artin algebra and modA the category of finitely generated right A-modules. Let M be an A-module and denote by addM the fill1 subcategory of modA consisting of the direct sums of direct summands of M . In [All M.Auslander considered the full subcategory Cy of modA consisting of the modules X having a presentation MI Mo -+ X -+ 0 wit,h Mt in addM, such that the induced sequence HomA(M, MI) -t HomA(M, Mo) -, HomA(hl, X) -+ 0 is exact, and proved that the restriction of the functor FM = HomA(hf, -) : modA -+ m o d E n d ~ ( M ) to Cp is full and faithful. T h w FM induces an equivalence between subcategories of modA and modEndA(M) respectively. More precisely, between Cy and h ( F M I c y ) , the image of the re~t~riction of the functor FM to C;f. When the module M is projective one getas the well known equivalence betweefl the subcategory of modA consisting of t,hc modules with a presentation in addM and modEndA(M). This paper is motivated by the above result. We start by defining for an A-module M and for every n > 0 full subcategories Cf of modA consisting of the modules X such that there is an exact sequence Mn -+ . . . -+ MI -, Mo

simply connected incidence algebras

It is known that the incidence algebra of a finite poset is not strongly simply connected if and only if its quiver contains a crown. We give a combinatorial condition on crowns which, if satisfied, forces the incidence algebra to be simply connected. The converse is not true, but we show that a simply connected incidence algebra which is not strongly simply connected always contains crowns satisfying this condition.

ARTIN RINGS WITH ALL IDEMPOTENT IDEALS PROJECTIVE

ABSTRACT. We study artin rings Λ with the property that all the idempotents two sided ideals of Λ are projective left Λ-modules. We give a characterization of these rings, and prove that their finitistic dimension is at most one. Using this result we study the Λ-modules of finite projective dimension.

Periodic modules over weakly symmetric algebras

where PrV1 + l e l + PO * C --) 0 is the start of a minimal projective resolution of C. We recall that a selfinjective algebra A is said to be weakly symmetric if for every indecomposable projective A -module PF P/x-P s sot P, where r denotes the radical of A and sot P denotes the socle of P, The main result of this paper is the following. If A is an indecr,mposaSle weakly symmetric algebra of infinite representation type, that is having an infinite number of indecomposable modules, and there is one indecomposable periodic A -module, then there are indecomposable periodic modules of arbitrarily large length, hence an infinite number of indecomposable periodic modules. We actually prove a more general statement, which also specialiies to the following result about hereditary algebra. If an hereditary algebra p4 has one indecomposable DTr-periodic A -module C, that is, (DTr)*C = C for some i 2 1, then it has indecomposable DTr-periodic modules of arbitrarily large length, hence an infinite number. Here D denotes the ordiqary duality for artin algebras and Tr the transpose (see [6] for definition). We apply the result about periodic modules over weakly symmetric algebras to show that if k is a field of characteristic p, a finite group such that p divides the order of G, and B is a block of infinite type in the group algebra kG, the:1 there are

C-filtered modules and proper costratifying systems

Abstract In this paper we define and study the notion of a proper costratifying system, which is a generalization of the so-called proper costandard modules to the context of stratifying systems. The proper costandard modules were defined by V. Dlab in his study of quasi-hereditary algebras (see Dlab, 1996 [D1] ).

Tilting Modules and the Subcategories (C M i )

In this article we further study the full subcategories of the category of finitely generated modules over an Artin algebra introduced in Platzeck and Pratti (2000), consisting of the modules having an add M resolution of length i, which remains exact under the functor Hom A (M, −). In particular, we characterize tilting modules in terms of these categories and determine when the transpose of a tilting module is a tilting module.

The grothendieck group of the category of modules of finite projective dimension over certain weakly triangular algebras

In this paper we study the category of finitely generated modules of finite projective dimension over a class of weakly triangular algebras, which includes the algebras whose idempotent ideals have finite projective dimension. In particular, we prove that the relations given by the (relative) almost split sequences generate the group of all relations for the Grothendieck group of P <∞(Λ) if and only if P <∞(Λ) is of finite type. A similar statement is known to hold for the category of all finitely generated modules over an artin algebra, and was proven by C.M.Butler and M. Auslander ( [B] and [A]).

Introduction to the Representation Theory of Finite-Dimensional Algebras: The Functorial Approach

These are the notes of a course given at the CIMPA School “Homological Methods, Representation Theory and Cluster Algebras,” Mar del Plata, Argentina, 2016. The aim of this brief course is to give an introduction to the functorial approach to the representation theory of finite-dimensional algebras, developed by Maurice Auslander and Idun Reiten, and is strongly based on the work “A functorial approach to representation theory,” by M. Auslander (Representations of Algebras. Springer, Berlin, 1981) [4].

Idempotent Ideals and the Igusa-Todorov Functions

Let Λ be an artin algebra and A