arXiv subjects

Edson Ribeiro Alvares

Publications and source records attributed to Edson Ribeiro Alvares.

6 recordsLinked to original sources

arXiv2026

Stratifying Systems via a Nested Family of Torsion Pairs

In this paper, we introduce the notion of a nested family of torsion pairs. We show that every stratifying system induces a nested family of torsion pairs and, conversely, that such a family together with a suitable object gives rise to a stratifying system; in particular, every stratifying system arises in this way. As an application, we present a stratifying system of infinite size that cannot be indexed by $(\mathbb{N}, \leq)$, where $\leq$ is the natural order.

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arXiv2025

On the global dimension three endomorphism algebras of the minimal generator-cogenerator

The main goal of this paper is to study the class of algebras for which the global dimension of the endomorphism ring of the generator-cogenerator, given by the sum of the projective and injective modules, is equal to three. We will refer to these algebras as representation-hereditary algebras. We show that these algebras are torsionless-finite, as defined by Ringel. These algebras do not necessarily have finite global dimension; however, when there is no non-zero morphism from an injective to a projective module, they have global dimension less than or equal to two, with some additional homological properties. By utilizing the general framework provided by the study of the representation dimension of an algebra, we present further homological consequences. In the case where these algebras are tame quasi-tilted algebras, we prove that they belong to certain classes of tilted algebras. Although not all tilted algebras are representation-hereditary, we provide sufficient conditions for them to be representation-hereditary.

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arXiv2017

(m,n)-Quasitilted and (m,n)-Almost Hereditary Algebras

Motivated by the study of (m,n)-quasitilted algebras, which are the piecewise hereditary algebras obtained from quasitilted algebras of global dimension two by a sequence of (co)tiltings involving n-1 tilting modules and m-1 cotilting modules, we introduce (m,n)-almost hereditary algebras. These are the algebras with global dimension m+n and such that any indecomposable module has projective dimension at most m, or else injective dimension at most n. We relate these two classes of algebras, among which (m,1)-almost hereditary ones play a special role. For these, we prove that any indecomposable module lies in the right part of the module category, or else in an m-analog of the left part. This is based on the more general study of algebras the module categories of which admit a torsion-free subcategory such that any indecomposable module lies in that subcategory, or else has injective dimension at most n.

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arXiv2017

Cohomology of partial smash products

We define the partial group cohomology as the right derived functor of the functor of partial invariants, we relate this cohomology with partial derivations and with the partial augmentation ideal and we show that there exists a Grothendieck spectral sequence relating cohomology of partial smash algebras with partial group cohomology and algebra cohomology.

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arXiv2017

The strong global dimension of piecewise hereditary algebras

Let T be a tilting object in a triangulated category equivalent to the bounded derived category of a hereditary abelian category with finite dimensional homomorphism spaces and split idempotents. This text investigates the strong global dimension, in the sense of Ringel, of the endomorphism algebra of T. This invariant is expressed using the infimum of the lengths of the sequences of tilting objects successively related by tilting mutations and where the last term is T and the endomorphism algebra of the first term is quasi-tilted. It is also expressed in terms of the hereditary abelian generating subcategories of the triangulated category.

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arXiv2016

Shapes of Auslander-Reiten Triangles

Our main theorem classifies the Auslander-Reiten triangles according to properties of the morphisms involved. As a consequence, we are able to compute the mapping cone of an irreducible morphism. We finish by showing a technique for constructing the connecting component of the derived category of any tilted algebra. In particular we obtain a technique for constructing the derived category of any tilted algebra of finite representation type.

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