arXiv subjects

Jose A. Velez-Marulanda

Publications and source records attributed to Jose A. Velez-Marulanda.

At least 19 recordsLinked to original sources

arXiv2026

On universal deformation rings and stable equivalences of Gorenstein-projective modules

Let $\mathbf{k}$ be a field and let $Λ$ and $Γ$ finite dimensional $\mathbf{k}$-algebras. Assume that ${_Γ}X_Λ$ and ${_Λ}Y_Γ$ are bimodules that define a singular equivalence of Morita type with level (in the sense of Z. Wang) between $Λ$ and $Γ$ and which also induce an equivalence between the stable categories of finitely generated Gorenstein-projective modules $Λ$-$\underline{\text{Gproj}}$ and $Γ$-$\underline{\text{Gproj}}$. We prove that if $V$ is an indecomposable object in $Λ$-$\underline{\text{Gproj}}$ with $\underline{\mathrm{End}}_Λ(V)\cong \mathbf{k}$, then $X\otimes_ΛV$ is an object in $Γ$-$\underline{\text{Gproj}}$ such that $\underline{\mathrm{End}}_Γ(X\otimes_ΛV)\cong \mathbf{k}$ and the universal deformation rings (in the sense of F.M. Bleher and the second author) $R(Λ,V)$ and $R(Γ, X\otimes_ΛV)$ are isomorphic. This result generalizes the one obtained by the second author assuming that $Λ$ and $Γ$ are Gorenstein $\mathbf{k}$-algebras.

math.RT↗

arXiv2025

On universal deformation rings and stable homogeneous tubes

Let $\mathbf{k}$ be a field of any characteristic and let $Λ$ be a finite dimensional $\mathbf{k}$-algebra. We prove that if $V$ is a finite dimensional right $Λ$-module that lies in the mouth of a stable homogeneous tube $\mathfrak{T}$ of the Auslander-Reiten quiver $Λ$ with $\underline{\mathrm{End}}_Λ(V)$ a division ring, then $V$ has a versal deformation ring $R(Λ,V)$ isomorphic to $\mathbf{k}[\![t]\!]$. As consequence we obtain that if $\mathbf{k}$ is algebraically closed, $Λ$ is a symmetric special biserial $\mathbf{k}$-algebra and $V$ is a band $Λ$-module with $\underline{\mathrm{End}}_Λ(V) \cong \mathbf{k}$ that lies in the mouth of its homogeneous tube, then $R(Λ,V)$ is universal and isomorphic to $\mathbf{k}[\![t]\!]$.

math.RT↗

arXiv2025

Exact weights and path metrics for triangulated categories and the derived category of persistence modules

We define exact weights on a triangulated category to be nonnegative functions on objects satisfying a subadditivity condition with respect to exact triangles. Such weights induce a metric on objects in the triangulated category, which we call a path metric. Our exact weights generalize the rank functions of J.\ Chuang and A.\ Lazarev and are analogous to the exact weights for an exact category given by the first author and J.\ Scott and D.\ Stanley. We show that cohomological functors from a triangulated category to an abelian category with an additive weight induce an exact weight on the triangulated category. We prove that triangle equivalences induce an isometry for the path metrics induced by cohomological functors. In the perfectly generated or compactly generated case, we use Brown representability to express the exact weight on the triangulated category. We give three characterizations of exactness for a weight on a triangulated category and show that they are equivalent. We also define Wasserstein distances for triangulated categories. Finally, we apply our work to derived categories of persistence modules and to representations of continuous quivers of type $\mathbb{A}$.

math.CT↗

arXiv2024

On Weak Universal Deformation Rings for Objects of EXT-FINITE Categories of Modules

Let $\A$ be a $\k$-algebra where $\k$ a field of arbitrary characteristic, and let $\mathscr{A}_\k$ be a full subcategory of $\A$-Mod, the abelian category of left $\A$-modules.Following M. Kleiner and I. Reiten, $\mathscr{A}_\k$ is {\it Hom-finite} if the hom-space between any two objects in $\mathscr{A}_\k$ is finite-dimensional over $\k$. We further say that $\mathscr{A}_\k$ is {\it Ext-finite} if $\dim_\k\Ext^i_\A(X,Y)<\infty$ for all objects $X$ and $Y$ in $\mathscr{A}_\k$. Let $V$ be an object in $\mathscr{A}_\k$. In this note we prove that if $\End_\A(V)$ is isomorphic to $\k$, then $V$ has a universal deformation ring $R(\A,V)$, which is a local complete Noetherian commutative $\k$-algebra whose residue field is also isomorphic to $\k$. We use this result to prove that if $\A$ is a local two-point infinite dimensional gentle $\k$-algebra (in the sense of V. Bekkert et al), then $R(\A,V)$ is isomorphic either to $\k$, to $\k[\![t]\!]/(t^2)$ or to $\k[\![t]\!]$.

math.RT↗

arXiv2024

On universal deformation rings of modules over a certain class of symmetric algebras of finite representation type

Let $\mathbf{k}$ be an algebraically closed field. Recently, K. Erdmann classified the symmetric $\mathbf{k}$-algebras $Λ$ of finite representation type such that every non-projective module $M$ has period dividing four. The goal of this paper is to determine the indecomposable modules $M$ over these class of algebras $Λ$ whose stable endomorphism ring is isomorphic to $\mathbf{k}$, and then calculate their corresponding universal deformation rings (in the sense of F. M. Bleher and the third author).

math.RT↗

arXiv2024

A short note on deformations of (strongly) Gorenstein-projective modules over the dual numbers

Let $\mathbf{k}$ be a field of arbitrary characteristic, and let $Λ$ be a finite dimensional $\mathbf{k}$-algebra. In this short note we prove that if $V$ is a finitely generated strongly Gorenstein-projective left $Λ$-module whose stable endomorphism ring $\underline{\mathrm{End}}_Λ(V)$ is isomorphic to $\mathbf{k}$, then $V$ has an universal deformation ring $R(Λ,V)$ isomorphic to the ring of dual numbers $\mathbf{k}[ε]$ with $ε^2=0$. As a consequence, we obtain the following result. Assume that $Q$ is a finite connected acyclic quiver, let $\mathbf{k} Q$ be the corresponding path algebra and let $Λ= \mathbf{k} Q[ε] = \mathbf{k} Q\otimes_{\mathbf{k}} \mathbf{k}[ε]$. If $V$ is a finitely generated Gorenstein-projective left $Λ$-module with $\underline{\mathrm{End}}_Λ(V)=\mathbf{k}$, then $V$ has an universal deformation ring $R(Λ,V)$ isomorphic to $\mathbf{k}[ε]

math.RT↗

arXiv2023

Singular equivalences of Morita type with level, Gorenstein algebras, and universal deformation rings

Let $\mathbf{k}$ be a field of arbitrary characteristic, let $Λ$ be a finite dimensional $\mathbf{k}$-algebra, and let $V$ be an indecomposable finitely generated non-projective Gorenstein-projective left $Λ$-module whose stable endomorphism ring is isomorphic to $\mathbf{k}$. In this article, we prove that the universal deformation rings $R(Λ,V)$ and $R(Λ,Ω_ΛV)$ are isomorphic, where $Ω_ΛV$ denotes the first syzygy of $V$ as a left $Λ$-module. We also prove the following result. Assume that $Λ$ is Gorenstein and that $Γ$ is another Gorenstein $\mathbf{k}$-algebra such that there exists $\ell \geq 0$ and a pair of bimodules $({_Γ}X_Λ, {_Λ}Y_Γ)$ that induces a singular equivalence of Morita type with level $\ell$ (as introduced by Z. Wang) between $Λ$ and $Γ$. Then the left $Γ$-module $X\otimes_ΛV$ is also Gorenstein-projective with stable endomorphism ring isomorphic to $\mathbf{k}$ and the universal deformation ring $R(Γ, X\otimes_ΛV)$ is isomorphic to $R(Λ, V)$.

math.RT↗

arXiv2019

Derived tame Nakayama algebras

We determine the derived representation type of Nakayama algebras and prove that a derived tame Nakayama algebra without simple projective module is gentle or derived equivalent to some skewed-gentle algebra, and as a consequence, we determine its singularity category.

math.RT↗

arXiv2019

On Deformations of Gorenstein-Projective Modules over Monomial Algebras with no Overlaps

Let $\mathbf{k}$ be a field of arbitrary characteristic, let $Λ$ be a finite dimensional $\mathbf{k}$-algebra, and let $V$ be an indecomposable Gorenstein-projective $Λ$-module with finite dimension over $\mathbf{k}$. It follows that $V$ has a well-defined versal deformation ring $R(Λ, V)$, which is complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$, and which is universal provided that the stable endomorphism ring of $V$ is isomorphic to $\mathbf{k}$. We prove that if $Λ$ is a monomial algebra without overlaps, then $R(Λ,V)$ is universal and isomorphic either to $\mathbf{k}$ or to $\mathbf{k}[[t]]/(t^2)$

math.RT↗

arXiv2019

On Deformations of Gorenstein-projective modules over Nakayama and triangular matrix algebras

Let $\mathbf{k}$ be a fixed field of arbitrary characteristic, and let $Λ$ be a finite dimensional $\mathbf{k}$-algebra. Assume that $V$ is a left $Λ$-module of finite dimension over $\mathbf{k}$. F. M. Bleher and the author previously proved that $V$ has a well-defined versal deformation ring $R(Λ,V)$ which is a local complete commutative Noetherian ring with residue field isomorphic to $\mathbf{k}$. Moreover, $R(Λ,V)$ is universal if the endomorphism ring of $V$ is isomorphic to $\mathbf{k}$. In this article we prove that if $Λ$ is a basic connected cycle Nakayama algebra without simple modules and $V$ is a Gorenstein-projective left $Λ$-module, then $R(Λ,V)$ is universal. Moreover, we also prove that the universal deformation rings $R(Λ,V)$ and $R(Λ, ΩV)$ are isomorphic, where $ΩV$ denotes the first syzygy of $V$. This result extends the one obtained by F. M. Bleher and D. J. Wackwitz concerning universal deformation rings of finitely generated modules over self-injective Nakayama algebras. In addition, we also prove the following result concerning versal deformation rings of finitely generated modules over triangular matrix finite dimensional algebras. Let $Σ=\begin{pmatrix} Λ& B\\0& Γ\end{pmatrix}$ be a triangular matrix finite dimensional Gorenstein $\mathbf{k}$-algebra with $Γ$ of finite global dimension and $B$ projective as a left $Λ$-module. If $\begin{pmatrix} V\\W\end{pmatrix}_f$ is a finitely generated Gorenstein-projective left $Σ$-module, then the versal deformation rings $R\left(Σ,\begin{pmatrix} V\\W\end{pmatrix}_f\right)$ and $R(Λ,V)$ are isomorphic.

math.RT↗

arXiv2017

Universal deformation rings for a class of self-injective special biserial algebras

Let $\mathbf{k}$ be an algebraically closed field of arbitrary characteristic, let $Λ$ be a finite dimensional $\mathbf{k}$-algebra and let $V$ be a $Λ$-module with stable endomorphism ring isomorphic to $\mathbf{k}$. If $Λ$ is self-injective, then $V$ has a universal deformation ring $R(Λ,V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$. Moreover, if $Λ$ is further a Frobenius $\mathbf{k}$-algebra, then $R(Λ,V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $Λ_{m,N}$-modules whose corresponding stable endomorphism ring is isomorphic to $\mathbf{k}$, and which lie either in a connected component of the stable Auslander-Reiten quiver of $Λ_{m,N}$ containing a module with endomorphism ring isomorphic to $\mathbf{k}$ or in a periodic component containing only string $Λ_{m,N}$-modules, where $m\geq 3$ and $N\geq 1$ are integers, and $Λ_{m,N}$ is a self-injective special biserial $\mathbf{k}$-algebra.

math.RT↗

arXiv2017

Deformations of complexes for finite dimensional algebras

Let $k$ be a field and let $Λ$ be a finite dimensional $k$-algebra. We prove that every bounded complex $V^\bullet$ of finitely generated $Λ$-modules has a well-defined versal deformation ring $R(Λ,V^\bullet)$ which is a complete local commutative Noetherian $k$-algebra with residue field $k$. We also prove that nice two-sided tilting complexes between $Λ$ and another finite dimensional $k$-algebra $Γ$ preserve these versal deformation rings. Additionally, we investigate stable equivalences of Morita type between self-injective algebras in this context. We apply these results to the derived equivalence classes of the members of a particular family of algebras of dihedral type that were introduced by Erdmann and shown by Holm to be not derived equivalent to any block of a group algebra.

math.RT↗

arXiv2017

On Singular Equivalences of Morita Type and Universal Deformation Rings for Gorenstein Algebras

Let $Λ$ be a finite-dimensional algebra over a fixed algebraically closed field $\mathbf{k}$ of arbitrary characteristic, and let $V$ be a finitely generated $Λ$-module. It follows from results previously obtained by F.M. Bleher and the third author that $V$ has a well-defined versal deformation ring $R(Λ, V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$. The third author also proved that if $Λ$ is a Gorenstein $\mathbf{k}$-algebra and $V$ is a Cohen-Macaulay $Λ$-module whose stable endomorphism ring is isomorphic to $\mathbf{k}$, then $R(Λ, V)$ is universal. In this article we prove that the isomorphism class of a versal deformation ring is preserved under singular equivalence of Morita type between Gorenstein $\mathbf{k}$-algebras.

math.RT↗

arXiv2017

On Universal Deformation Rings for Gorenstein Algebras

Let $\mathbf{k}$ be an algebraically closed field, and let $Λ$ be a finite dimensional $\mathbf{k}$-algebra. We prove that if $Λ$ is a Gorenstein algebra, then every finitely generated Cohen-Macaulay $Λ$-module $V$ whose stable endomorphism ring is isomorphic to $\mathbf{k}$ has a universal deformation ring $R(Λ,V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$, and which is also stable under taking syzygies. We investigate a particular non-self-injective Gorenstein algebra $Λ_0$, which is of infinite global dimension and which has exactly three isomorphism classes of finitely generated indecomposable Cohen-Macaulay $Λ_0$-modules $V$ whose stable endomorphism ring is isomorphic to $\mathbf{k}$. We prove that in this situation, $R(Λ_0,V)$ is isomorphic either to $\mathbf{k}$ or to $\mathbf{k}[[t]]/(t^2)$.

math.RT↗

arXiv2017

Universal Deformation Rings for Complexes over Finite-Dimensional Algebras

Let $\mathbf{k}$ be field of arbitrary characteristic and let $Λ$ be a finite dimensional $\mathbf{k}$-algebra. From results previously obtained by F.M Bleher and the author, it follows that if $V^\bullet$ is an object of the bounded derived category $\mathcal{D}^b(Λ\textup{-mod})$ of $Λ$, then $V^\bullet$ has a well-defined versal deformation ring $R(Λ, V^\bullet)$, which is complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$, and which is universal provided that $\textup{Hom}_{\mathcal{D}^b(Λ\textup{-mod})}(V^\bullet, V^\bullet)=\mathbf{k}$. Let $\mathcal{D}_\textup{sg}(Λ\textup{-mod})$ denote the singularity category of $Λ$ and assume that $V^\bullet$ is a bounded complex whose terms are all finitely generated Gorenstein projective left $Λ$-modules. In this article we prove that if $\textup{Hom}_{\mathcal{D}_\textup{sg}(Λ\textup{-mod})}(V^\bullet, V^\bullet)=\mathbf{k}$, then the versal deformation ring $R(Λ, V^\bullet)$ is universal. We also prove that certain singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism class of versal deformation rings of bounded complexes whose terms are finitely generated Gorenstein projective $Λ$-modules.

math.RT↗

arXiv2014

On the Infinitude of Prime Ideals in Dedekind Domains

Let $R$ be an infinite Dedekind domain with at most finitely many units, and let $K$ denote its field of fractions. We prove the following statement. If $L/K$ is a finite Galois extension of fields and $\mathcal{O}$ is the integral closure of $R$ in $L$, then $\mathcal{O}$ contains infinitely many prime ideals. In particular, if $\mathcal{O}$ is further a unique factorization domain, then $\mathcal{O}$ contains infinitely many non-associate prime elements.

math.NT↗

arXiv2012

Universal deformation rings of string modules over a certain symmetric special biserial algebra

Let $\k$ be an algebraically closed field, let $\A$ be a finite dimensional $\k$-algebra and let $V$ be a $\A$-module with stable endomorphism ring isomorphic to $\k$. If $\A$ is self-injective then $V$ has a universal deformation ring $R(\A,V)$, which is a complete local commutative Noetherian $\k$-algebra with residue field $\k$. Moreover, if $Λ$ is also a Frobenius $\k$-algebra then $R(\A,V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $\Ar$-modules whose stable endomorphism ring isomorphic to $\k$, where $\Ar$ is a symmetric special biserial $\k$-algebra that has quiver with relations depending on the four parameters $ \bar{r}=(r_0,r_1,r_2,k)$ with $r_0,r_1,r_2\geq 2$ and $k\geq 1$.

math.RT↗

arXiv2011

Universal deformation rings of modules over Frobenius algebras

Let $k$ be a field, and let $Λ$ be a finite dimensional $k$-algebra. We prove that if $Λ$ is a self-injective algebra, then every finitely generated $Λ$-module $V$ whose stable endomorphism ring is isomorphic to $k$ has a universal deformation ring $R(Λ,V)$ which is a complete local commutative Noetherian $k$-algebra with residue field $k$. If $Λ$ is also a Frobenius algebra, we show that $R(Λ,V)$ is stable under taking syzygies. We investigate a particular Frobenius algebra $Λ_0$ of dihedral type, as introduced by Erdmann, and we determine $R(Λ_0,V)$ for every finitely generated $Λ_0$-module $V$ whose stable endomorphism ring is isomorphic to $k$.

math.RT↗