Flávio U. Coelho

Two-sided gluings of tilted algebras

We study the class of algebras A satisfying the property: all but at most finitely many non-isomorphic indecomposable A-modules are such that all their predecessors have projective dimension at most one, or all their successors have injective dimension at most one. Such a class includes the tilted algebras [D. Happel, C. Ringel, Trans. Amer. Math. Soc. 274 (1982) 399–443], the quasi-tilted algebras [D. Happel, I. Reiten, S. Smalo, Mem. Am. Math. Soc. 120 (1996) 575], the shod algebras [F.U. Coelho, M. Lanzilotta, Manuscripta Mathematica 100 (1999) 1–11], the weakly shod [F.U. Coelho, M. Lanzilotta, Preprint, 2001], and the left and right glued algebras [I. Assem, F.U. Coelho, J. Pure Appl. Algebra 96 (3) (1994) 225–243].

Weakly shod algebras

Abstract We study the class of algebras having a path of nonisomorphisms from an indecomposable injective module to an indecomposable projective module and any such path is bounded by a fixed number. We show that such an algebra is an iteration of one-point extensions starting at a product of tilted algebras. This allows us to describe, for instance, its Auslander–Reiten quiver.

Glueings of tilted algebras

Abstract Let A be a basic connected finite-dimensional algebra over an algebraically closed field. We show that if A has all its indecomposable projectives (or injectives) lying in a component of the Auslander—Reiten quiver consisting entirely of postprojective (or preinjective, respectively) modules in the sense of Auslander and Smalo then A is a finite enlargement in the postprojective (or preinjective, respectively) components of a finite set of tilted algebras having complete slices in these components. We call such an algebra A a left (or right, respectively) glued algebra and study some of its homological properties in particular in the case where A is itself a tilted algebra.

Module categories with infinite radical square zero are of finite type

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.

Infinitely generated complements to partial tilting modules

We study the existence of complements to a partial tilting module over an arbitrary ring. As a consequence, we show that a finitely generated partial tilting module over an artin algebra has a (possibly infinitely generated) complement.

ENDOMORPHISM ALGEBRAS OF PROJECTIVE MODULES OVER LAURA ALGEBRAS

Let A be a connected artin algebra, and e be an idempotent in A such that B=eAe is connected. We show here that if A is laura, left (or right) glued or weakly shod, so is B, respectively. Our proof yields also similar (and known) results for shod and quasi-tilted algebras.

Simply connected tame quasi-tilted algebras

Abstract We show that a tame quasi-tilted algebra A (over an algebraically closed field) is simply connected if and only if its first Hochschild cohomology group (with coefficients in the biomodule AAA) vanishes. We also classify these algebras, and give characterisations of those tame quasi-tilted algebras which are strongly simply connected.

THE BOUND QUIVER OF A SPLIT EXTENSION

In this paper, we give a sufficient (which is also necessary under a compatibility hypothesis) condition on a set of arrows in the quiver of an algebra A so that A is a split extension of A/M, where M is the ideal of A generated by the classes of these arrows. We also compare the notion of split extension with that of semiconvex extension of algebras.

Tilt-Critical Algebras of Tame Type

Let A be a finite dimensional k-algebra over an algebraically closed field. Assume A = kQ/I where Q is a quiver without oriented cycles. We say that A is tilt-critical if it is not tilted but every proper convex subcategory of A is tilted. We describe the tilt-critical algebras which are strongly simply connected and tame.

Auslander generators of iterated tilted algebras

Let A be an iterated tilted algebra. We will construct an Auslander generator M in order to show that the representation dimension of A is three in case A is representation infinite.