Raymundo Bautista

Almost split sequences for relatively projective modules

The paper deals with almost split sequences. Introduced in [2] for the category mod A of finitely generated modules over an artin algebra A, almost split sequences were later found in the category of lattices over an order [l, 43, as well as in certain subcategories of mod A [6, l&3]. It is generally recognized that if almost split sequences exist, the subcategory has nice properties. We are concerned with the subcategory of relatively projective modules. Let R be a field or a Dedekind domain with the field of quotients k, and let A and A be finite-dimensional R-algebras or R-orders, respectively, with A mapped into A via an R-algebra map i: A -+ A. Here we understand orders and lattices in the sense of [ 1, p. 85, Example (b)]. Namely, A is an R-order if it is a noetherian R-algebra projective as an R-module, and x = k OR A is a self-injective ring. A-mod denotes the category of finitely generated left A-modules if R is a field, or the category of left A-lattices if R is a Dedekind domain, where a left A-module h4 is a lattice if it is a finitely generated projective R-module such that k Ox M is a projective

On irreducible maps

The notion of irreducible map was introduced by M. Auslander and I. Reiten in [3] and plays an important role in the representation theory of artin algebras. We recall that an artin ring A is said to be an artin algebra if its center C is an artin ring and A is finitely generated left A-module. Now choose a complete set P x ,. .. , P s of representatives of the isomorphism classes of indecomposable projectives in mod(A), we will denote by pr A the full subcategory of mod A whose objects are P t ,. .. , P s. A map g: X — • Y in mod(A) is said to be irre-ducible if g is neither a split monomorphism nor a split epimorphism and for any commutative diagram X~^+Y f\/h Z ƒ is a splittable monomorphism or ft is a splittable epimorphism. We study irreducible maps in mod(A) by using properties of the Jacobson radical of mod(A). We recall that the Jacobson radical of mod(A) is the subfunc-tor rad of the two variable functor Hom: (mod(A)) op x mod(A) — • Ab defined by rad(X, Y) = {ƒ G Hom(X, Y)\ 1-gf is invertible for every g G Hom(7, X)} = {ƒ G Hom(X, F)I 1-fli is invertible for any ft G Hom(Z, Y)}. It is easy to prove that if X and Y are indécomposables, then rad(X, Y) consists of all nonisomorphisms, from X to Y. We can prove the following result: PROPOSITION 1. Let C and D be indécomposables in mod(A). Then (a) Amapf\C—* is irreducible iff f G rad(C, D) and f

Representations of a k-algebra over the rational functions over k

rad 2 (C, D), where rad 2 (C, D) consists of all maps of the form t x t 2 with t 2 € rad(C, X) and t x G rad(X, D).

On restrictions of generic modules of tame algebras

Abstract For Λ a finite-dimensional k-algebra, k a field, we study the relations between the category of all left Λ-modules, ModΛ, and the category of finitely generated Λ⊗kk(x)=Λk(x)-modules, modΛk(x). In particular, we consider those G∈modΛk(x) such that ΛG is indecomposable. We prove that such modules are in the mouth of components in modΛk(x) which are tubes or have the shape Z A ∞ , Z D ∞ , or Z B ∞ .

On Hom-spaces of tame algebras

Given a convex algebra ∧0 in the tame finite-dimensional basic algebra ∧, over an algebraically closed field, we describe a special type of restriction of the generic ∧-modules.

On Modules and Complexes Without Self-Extensions

Let Λ be a finite dimensional algebra over an algebraically closed field k and Λ has tame representation type. In this paper, the structure of Hom-spaces of all pairs of indecomposable Λ-modules having dimension smaller than or equal to a fixed natural number is described, and their dimensions are calculated in terms of a finite number of finitely generated Λ-modules and generic Λ-modules. In particular, such spaces are essentially controlled by those of the corresponding generic modules.

On Rigid Minimal Presentations

Let Λ be an Artin algebra over a commutative Artinian ring, k. If M is a finitely generated left Λ -module, we denote by Ω (M) the kernel of η M : P M → M a minimal projective cover. We prove that if M and N are finitely generated left Λ -modules and Ext Λ 1 (M, M) = 0, Ext Λ 1 (N, N) = 0, then M≅ N if and only if M/rad M≅ N/rad N and Ω (M)≅ Ω (N). Now if k is an algebraically closed field and (d i ) i∈ℤ is a sequence of nonnegative integers almost all of them zero, then we prove that the family of objects X ∈ 𝒟 b (Λ), the bounded derived category of Λ, with Hom𝒟 b (Λ)(X,X[1]) = 0 and dim k H i (X) = d i for all i ∈ ℤ, has only a finite number of isomorphism classes (see Huisgen-Zimmermann and Saorín, 2001).

On Some Tame and Discrete Families of Modules

Abstract Let Λ be a basic finite dimensional k-algebra which contains a semisimple algebra S separable over k with Λ = S ⊕ J, where J is the Jacobson radical of Λ. The category 𝒫(Λ) of radical morphisms between projective left Λ-modules has an exact structure. For ϕ1, ϕ2 in 𝒫(Λ), denote by Ext𝒫(ϕ1, ϕ2) the corresponding extension group. A finitely generated left Λ-module M has rigid minimal projective presentation if Ext𝒫(ϕ, ϕ) = 0. We will see that if M is indecomposable and has rigid minimal projective presentation then EndΛ(M)/radEndΛ(M) ≅ EndΛ(T) for T a simple Λ-module.

A wild bocs having strong homogeneous property

Here we consider algebras Λ over an algebraically closed field k, which are k-finite dimensional.

A MATRIX DESCRIPTION OF A WILD CATEGORY

Bocs, which is the abbreviated form of bimodule over a categary with coalgebra structure, was introduced by Kleiner and Rojter in 1975 and developed by Drozd in 1979, then formulated by Crawley-Boevey in 1988. Let k be an algebraically closed field, A a finitely dimensional k-algebra. Then there exists a bocs B over k associated to A. From this relation Drozd proved one of the most important theorems in representation theory of algebra, namely, a finitely dimensional k-algebra is either of representation tame type or of representation wild type, but