arXiv subjects

Teresa Conde

Publications and source records attributed to Teresa Conde.

10 recordsLinked to original sources

arXiv2026

Comparing self-dual corings, Frobenius corings and Ringel self-duality

Self-dual corings and self-dual algebra extensions are classified and these (self-)dualities are compared with Ringel (self-)duality, revealing fundamental differences. To each coring $\mathcal C$, two algebras are associated, known as the left and the right dual algebra of $\mathcal C$. It is shown that these two algebras coincide in a natural way if and only if $\mathcal C$ is a Frobenius coring. Various other equivalent characterisations of $\mathcal C$ being self-dual are given, in terms of certain algebra extensions being Frobenius extensions and in terms of certain forgetful or restriction functors being Frobenius functors. Ringel self-duality however is shown to be rather different, for which a homological explanation is given.

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arXiv2026

Triangular decompositions: Reedy algebras and quasi-hereditary algebras

Finite-dimensional Reedy algebras form a ring-theoretic analogue of Reedy categories and were recently proved to be quasi-hereditary. We identify Reedy algebras with quasi-hereditary algebras admitting a triangular (or Poincaré-Birkhoff-Witt type) decomposition into the tensor product of two oppositely directed subalgebras over a common semisimple subalgebra. This exhibits homological and representation-theoretic structure of the ingredients of the Reedy decomposition and it allows to give a characterisation of Reedy algebras in terms of idempotent ideals occurring in heredity chains, providing an analogue for Reedy algebras of a result of Dlab and Ringel on quasi-hereditary algebras.

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arXiv2026

Exact Borel subalgebras, idempotent quotients and idempotent subalgebras

This article studies the compatibility of Koenig's notion of an exact Borel subalgebra of a quasi-hereditary or, more generally, standardly stratified algebra with taking idempotent subalgebras or quotients. As an application, we provide bounds for the multiplicities of indecomposable projectives in the principal blocks of BGG category $\mathcal{O}$ having basic regular exact Borel subalgebras.

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arXiv2023

A functorial approach to rank functions on triangulated categories

We study rank functions on a triangulated category $\mathcal{C}$ via its abelianisation $\operatorname{mod}\mathcal{C}$. We prove that every rank function on $\mathcal{C}$ can be interpreted as an additive function on $\operatorname{mod}\mathcal{C}$. As a consequence, every integral rank function has a unique decomposition into irreducible ones. Furthermore, we relate integral rank functions to a number of important concepts in the functor category $\operatorname{Mod}\mathcal{C}$. We study the connection between rank functions and functors from $\mathcal{C}$ to locally finite triangulated categories, generalising results by Chuang and Lazarev. In the special case $\mathcal{C}=\mathcal{T}^c$ for a compactly generated triangulated category $\mathcal{T}$, this connection becomes particularly nice, providing a link between rank functions on $\mathcal{C}$ and smashing localisations of $\mathcal{T}$. In this context, any integral rank function can be described using the composition length with respect to certain endofinite objects in $\mathcal{T}$. Finally, if $\mathcal{C}=\operatorname{per}(A)$ for a differential graded algebra $A$, we classify homological epimorphisms $A\to B$ with $\operatorname{per}(B)$ locally finite via special rank functions which we call idempotent.

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arXiv2021

All quasihereditary algebras with a regular exact Borel subalgebra

Not every quasihereditary algebra $(A,Φ,\unlhd)$ has an exact Borel subalgebra. A theorem by Koenig, Külshammer and Ovsienko asserts that there always exists a quasihereditary algebra Morita equivalent to $A$ that has a regular exact Borel subalgebra, but a characterisation of such a Morita representative is not directly obtainable from their work. This paper gives a criterion to decide whether a quasihereditary algebra contains a regular exact Borel subalgebra and provides a method to compute all the representatives of $A$ that have a regular exact Borel subalgebra. It is shown that the Cartan matrix of a regular exact Borel subalgebra of a quasihereditary algebra $(A,Φ,\unlhd)$ only depends on the composition factors of the standard and costandard $A$-modules and on the dimension of the $\operatorname{Hom}$-spaces between standard $A$-modules. We also characterise the basic quasihereditary algebras that admit a regular exact Borel subalgebra.

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arXiv2021

All exact Borel subalgebras and all directed bocses are normal

Recently, Brzeziński, Koenig and Külshammer have introduced the notion of normal exact Borel subalgebra of a quasihereditary algebra. They have shown that there exists a one-to-one correspondence between normal directed bocses and quasihereditary algebras with a normal and homological exact Borel subalgebra. In this short note, we prove that every exact Borel subalgebra is automatically normal. As a corollary, we conclude that every directed bocs has a group-like element. These results simplify Brzeziński, Koenig and Külshammer's bijection.

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arXiv2020

Gabriel-Roiter measure, representation dimension and rejective chains

The Gabriel-Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel-Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel-Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel-Roiter measure $μ$ in an abelian length category $\mathcal{A}$, there exists an object $X'$ which depends on $X$ and $μ$, such that $Γ= \operatorname{End}_{\mathcal{A}}(X \oplus X')$ has finite global dimension. Analogously to Iyama's original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.

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arXiv2017

$Δ$-filtrations and projective resolutions for the Auslander-Dlab-Ringel algebra

The ADR algebra $R_A$ of an Artin algebra $A$ is a right ultra strongly quasihereditary algebra (RUSQ algebra). In this paper we study the $Δ$-filtrations of modules over RUSQ algebras and determine the projective covers of a certain class of $R_A$-modules. As an application, we give a counterexample to a claim by Auslander-Platzeck-Todorov, concerning projective resolutions over the ADR algebra.

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arXiv2017

The Ringel dual of the Auslander-Dlab-Ringel algebra

The ADR algebra $R_A$ of a finite-dimensional algebra $A$ is a quasihereditary algebra. In this paper we study the Ringel dual $\mathcal{R}(R_A)$ of $R_A$. We prove that $\mathcal{R}(R_A)$ can be identified with $(R_{A^{op}})^{op}$, under certain 'minimal' regularity conditions for $A$. We also give necessary and sufficient conditions for the ADR algebra to be Ringel selfdual.

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arXiv2016

The quasihereditary structure of the Auslander-Dlab-Ringel algebra

Given an arbitrary algebra $A$ we may associate to it a special endomorphism algebra, $R_A$, introduced by Auslander. Dlab and Ringel constructed a heredity chain for $R_A$, proving that every algebra $A$ has an associated highest weight theory. In this paper we investigate the quasihereditary structure of $R_A$ using an axiomatic approach.

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