arXiv subjects

Hernán Giraldo

Publications and source records attributed to Hernán Giraldo.

9 recordsLinked to original sources

arXiv2026

Reflections and Sheafifications in Algebraic and Topological Categories

In this article, we develop an explicit categorical realization of sheafification based on colimits, products, and subobjects, emphasizing its behavior in algebraic and topological-algebraic settings. We prove that if $\mathcal{C}$ is a reflective subcategory of a category $\mathcal{A}$, then the presheaf category $\mathbf{Psh}(X,\mathcal{C})$ is reflective in $\mathbf{Psh}(X,\mathcal{A})$. We further investigate the interaction between reflections and sheafification, obtaining natural conditions under which these constructions are naturally isomorphic.

math.GN↗

arXiv2023

A partial classification of simple regular representations of bimodules type $(2,\,2)$ over $\mathbb{C}(\!(\varepsilon)\!)$

In this paper, we use Galois descent techniques to find suitable representatives of the regular simple representations of the species of type $(2,2)$ over $k_n := k[\varepsilon^{1/n}]$, where $n$ is a positive integer and $k:=\mathbb{C}(\!(\varepsilon)\!)$ is the field of Laurent series over the complexes. These regular representations are essential for the definition of canonical algebras. Our work is inspired by the work done for species of type $(1,4)$ on $k$ in ``A model for the canonical algebras of bimodules type (1, 4) over truncated polynomial rings''. We presents all the regular simple representations on the $n$-crown quiver, and from these, we establish a partial classification of regular simple representations of bimodules type $(2,2)$.

math.RT↗

arXiv2023

Relations Between the Strong Global dimension, Complexes of Fixed Size and Derived Category

Let $\mathbb{Z}$ be the integer numbers, $\mathbb{k}$ an algebraically closed field, $Λ$ a finite dimensional $\mathbb{k}$-algebra, mod$Λ$ the category of finitely generated right modules, proj$Λ$ the full subcategory of mod$Λ$ consisting of all projective $Λ$-modules, and $C_n(projΛ)$ the bounded complexes of projective $Λ$-modules of fixed size for any integer $n\geq2$. We found an algorithm to calculate the strong global dimension of $Λ$, when $Λ$ has finite strong global dimension and derived-discrete, using the Auslander-Reiten quivers of the categories $C_n(projΛ)$. Also, we show the relation between the Auslander-Reiten quiver of the bounded derived category $D^b(Λ)$ and the Auslander-Reiten quiver of $C_{η+1}(projΛ)$, where $η=s.gl.dim(Λ)$ (strong global dimension of $Λ$).

math.RT↗

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 $\mathbb{P}^1_{\mathbf{k}}$

math.RT↗

arXiv2020

String and band complexes over SAG algebras

We give a combinatorial description of a family of indecomposable objects in the bounded derived categories of a new class of algebras: string almost gentle algebras. These indecomposable objects are, up to isomorphism, the string and band complexes introduced by V. Bekkert and H. Merklen. With this description, we give a necessary and sufficient condition for a given string complex to have infinite minimal projective resolution and we extend this condition for the case of string algebras. Using this characterization we establish a sufficient condition for a string almost gentle algebra (or a string algebra) to have infinite global dimension.

math.RT↗

arXiv2020

Periodic String Complexes over String Algebras

In this paper we develop combinatorial techniques for the case of string algebras with the aim to give a characterization of string complexes with infinite minimal projective resolution. These complexes will be called \textit{periodic string complexes}. As a consequence of this characterization, we give two important applications. The first one, is a sufficient condition for a string algebra to have infinite global dimension. In the second one, we exhibit a class of indecomposable objects in the derived category for a special case of string algebras. Every construction, concept and consequence in this paper is followed by some illustrative examples.

math.RT↗

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.

math.RT↗

arXiv2016

Differential Galois Groups and Representation of Quivers for Seismic Models with Constant Hessian of Square of Slowness

The trajectory of energy is modeled by the solution of the Eikonal equation, which can be solved by solving a Hamiltonian system. This system is amenable of treatment from the point view of the theory of Differential Algebra. In particular, by Morales-Ramis theory it is possible to analyze integrable Hamiltonian systems through the abelian structure of their variational equations. In this paper we obtain the abelian differential Galois groups and the representation of the quiver, that allow us to obtain such abelian differential Galois groups, for some seismic models with constant Hessian of square of slowness, proposed in Cerveny 2001, which are equivalent to linear Hamiltonian systems with three uncoupled harmonic oscillators.

math-ph↗

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.

math.RT↗