arXiv subjects

Sunny Roy

Publications and source records attributed to Sunny Roy.

4 recordsLinked to original sources

arXiv2025

Exact structures and maximal canonically Jordan recoverable subcategories for modules over type $A$ algebras

On one hand, exact structures were introduced by D. Quillen in the '70s. They can be defined as collections of short exact sequences in a fixed abelian category satisfying additional properties. On the other hand, in a recent work, A. Garver, R. Patrias, and H. Thomas introduced Jordan recoverability. Given a bounded quiver $(Q,R)$, a full additive subcategory of $\operatorname{rep}(Q,R)$ is said to be Jordan recoverable if any $X \in \mathscr{C}$ can be recovered, up to isomorphism, from the Jordan form of its generic nilpotent endomorphisms. Such a subcategory $\mathscr{C}$ is said to be canonically Jordan recoverable if, moreover, there exists a precise algebraic procedure that allows one to get back $X \in \mathscr{C}$ from that same Jordan form data. We introduce a new family of operators, called Gen-Sub operators $\operatorname{GS}_\mathcal{E}$, parametrized by the exact structures $\mathcal{E}$ of abelian categories. After showing some properties of those operators in hereditary abelian categories, by focusing on the setting of modules over path algebras of type $A$ quivers endowed with the diamond exact structure $\mathcal{E}_\diamond$, we establish that the maximal canonically Jordan recoverable subcategories are precisely of the form $\operatorname{GS}_{\mathcal{E}_\diamond}(T)$ for some tilting objects $T$.

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arXiv2024

An exact structure approach to almost rigid modules over quivers of type $\mathbb{A}$

Let $A$ be the path algebra of a quiver of Dynkin type $\mathbb{A}_n$. The module category $\text{mod}\,A$ has a combinatorial model as the category of diagonals in a polygon $S$ with $n+1$ vertices. The recently introduced notion of almost rigid modules is a weakening of the classical notion of rigid modules. The importance of this new notion stems from the fact that maximal almost rigid $A$-modules are in bijection with the triangulations of the polygon $S.$ In this article, we give a different realization of maximal almost rigid modules. We introduce a non-standard exact structure $\mathcal{E}_\diamond$ on $\text{mod}\,A$ such that the maximal almost rigid $A$-modules in the usual exact structure are exactly the maximal rigid $A$-modules in the new exact structure. A maximal rigid module in this setting is the same as a tilting module. Thus the tilting theory relative to the exact structure $\mathcal{E}_\diamond$ translates into a theory of maximal almost rigid modules in the usual exact structure. As an application, we show that with the exact structure $\mathcal{E}_\diamond$, the module category becomes a 0-Auslander category in the sense of Gorsky, Nakaoka and Palu. We also discuss generalizations to quivers of type $\mathbb{D}$ and gentle algebras.

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arXiv2019

Admissible intersection and sum property

We introduce subclasses of exact categories in terms of admissible intersections or admissible sums or both at the same time. These categories are recently studied by Br\"ustle, Hassoun, Shah, Tattar and Wegner to give characterisations of quasi-abelian and abelian categories. We also generalise the Schur lemma to the context of exact categories.

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arXiv2018

Reduction of exact structures

Examples of exact categories in representation theory are given by the category of Delta-filtered modules over quasi-hereditary algebras, but also by various categories related to matrix problems, such as poset representations or representations of bocses. Motivated by the matrix problem background, we study in this article the reduction of exact structures, and consider the poset Ex(A) of all exact structures on a fixed additive category A. This poset turns out to be a complete lattice, and under suitable conditions results of Enomoto's imply that it is boolean. We initiate in this article a detailed study of exact structures E by generalizing notions from abelian categories such as the length of an object relative to E and the quiver of an exact category (A,E). We investigate the Gabriel-Roiter measure for (A,E), and further study how these notions change when the exact structure varies.

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