arXiv subjects

José A. Vélez-Marulanda

Publications and source records attributed to José A. Vélez-Marulanda.

5 recordsLinked to original sources

arXiv2025

Universal deformation rings of a special class of modules over generalized Brauer tree algebras

Let $\Bbbk$ be an algebraically closed field and $Λ$ a generalized Brauer tree algebra over $\Bbbk$. We compute the universal deformation rings of the periodic string modules over $Λ$. Moreover, for a specific class of generalized Brauer tree algebras $Λ(n,\overline{m})$, we classify the universal deformation rings of the modules lying in $Ω$-stable components $\mathfrak{C}$ of the stable Auslander-Reiten quiver provided that $\mathfrak{C}$ contains at least one simple module. Our approach uses several tools and techniques from the representation theory of Brauer graph algebras. Notably, we leverage Duffield's work on the Auslander-Reiten theory of these algebras and Opper-Zvonareva's results on derived equivalences between Brauer graph algebras.

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arXiv2024

Liftings of point-wise finite dimensional persistence modules over local commutative Artinian rings

Let $\mathbf{k}$ be a field and let $V: \mathscr{C} \to \mathbf{k}\textup{-Mod}$ be a point-wise finite dimensional persistence modules, where $\mathscr{C}$ is a small category. Assume that for all local Artinian $\mathbf{k}$-algebras $R$ with residue field isomorphic to $\mathbf{k}$, there is a generalized persistence module $M: \mathscr{C} \to R\textup{-Mod}$, such that for all $x\in \mathrm{Ob}(\mathscr{C})$, $M(x)$ is free over $R$ with finite rank and $\mathbf{k}\otimes_R M(x)\cong V(x)$. If $V$ is a direct sum of indecomposable persistence modules $V_I: \mathscr{C}\to \mathbf{k}\textup{-Mod}$ with endomorphism ring isomorphic to $\mathbf{k}$, then $M$ is a direct sum of indecomposables $M_I:\mathscr{C}\to R\textup{-Mod}$ with endomorphism ring isomorphic to $R$

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arXiv2021

On Auslander-Reiten components of string complexes for a certain class of symmetric special biserial algebras

Let $\mathbf{k}$ be an algebraically closed field. In this article, inspired by the description of indecomposable objects in the derived category of a gentle algebra obtained by V. Bekkert and H. A. Merklen, we define string complexes for a certain class $\mathscr{C}$ of symmetric special biserial algebras, which are indecomposable perfect complexes in the corresponding derived category. We also prove that if $Λ$ is a $\mathbf{k}$-algebra in the class $\mathscr{C}$ and $P^\bullet$ is a string complex over $Λ$, then $P^\bullet$ lies in the rim of its Auslander-Reiten component.

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arXiv2021

On a deformation theory of finite dimensional modules over repetitive algebras

Let $Λ$ be a basic finite dimensional algebra over an algebraically closed field $\mathbf{k}$, and let $\widehatΛ$ be the repetitive algebra of $Λ$. In this article, we prove that if $\widehat{V}$ is a left $\widehatΛ$-module with finite dimension over $\mathbf{k}$, then $\widehat{V}$ has a well-defined versal deformation ring $R(\widehatΛ,\widehat{V})$, which is a local complete Noetherian commutative $\mathbf{k}$-algebra whose residue field is also isomorphic to $\mathbf{k}$. We also prove that $R(\widehatΛ,\widehat{V})$ is universal provided that $\underline{\mathrm{End}}_{\widehatΛ}(\widehat{V})=\mathbf{k}$ and that in this situation, $R(\widehatΛ,\widehat{V})$ is stable after taking syzygies. We apply the obtained results to finite dimensional modules over the repetitive algebra of the $2$-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over $\mathbb{P}^1_{\mathbf{k}}$

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arXiv2019

On irreducible morphisms and Auslander-Reiten triangles in the stable category of modules over repetitive algebras

Let $\mathbf{k}$ be an algebraically closed field, let $Λ$ be a finite dimensional $\mathbf{k}$-algebra, and let $\widehatΛ$ be the repetitive algebra of $Λ$. For the stable category of finitely generated left $\widehatΛ$-modules $\widehatΛ$-\underline{mod}, we show that the irreducible morphisms fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. We next use this fact in order to describe the shape of the Auslander-Reiten triangles in $\widehatΛ$-\underline{mod}. We use the fact (and prove) that every Auslander-Reiten triangle in $\widehatΛ$-\underline{mod} is induced from an Auslander-Reiten sequence of finitely generated left $\widehatΛ$-modules.

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