arXiv subjects

Pedro Rizzo

Publications and source records attributed to Pedro Rizzo.

At least 19 recordsLinked to original sources

arXiv2026

The Structure of $C^\infty$-Superschemes

This paper establishes a structural generalization of Batchelor's theorem within the framework of $C^\infty$-superschemes. Our main result proves that any Batchelor space satisfies a global splitness condition, establishing an isomorphism between the structure sheaf and its associated graded sheaf. Although this isomorphism is non-canonical, the existence of a splitting endows the structure sheaf with a natural $\mathbb{Z}_{\geq 0}$-grading. This grading is shown to be equivalent to the data of an even superderivation, which we term an Euler vector field. Consequently, global splittings of $C^\infty$-superspaces can be characterized in terms of Euler vector fields, providing a differential-geometric formulation of the splitting.

math.AG↗

arXiv2025

Continuous Linear Series

We parameterize by a fine moduli space all degenerations of linear series to a singular curve which is the union of two smooth components meeting transversally at a single point. For this we introduce a novel object in the study of degenerations of linear series, which is the continuous linear series. Our moduli space can be regarded as a Hilbert quotient, in the terminology introduced by Kapranov, and is a new compactification of Osserman moduli space of exact limit linear series, and consequently, of Eisenbud and Harris moduli space of refined limit linear series on the curve.

math.AG↗

arXiv2025

$C^\infty$-superrings and $C^\infty$-superschemes

This paper develops a theory of $C^\infty$-superrings and their associated $C^\infty$-superschemes. We prove a key equivalence between the category of fair affine $C^\infty$-superschemes and the category of fair $C^\infty$-superrings. We place special emphasis on split $C^\infty$-superrings, which generalize the function algebras of supermanifolds and serve as building blocks for more complex, non-split structures.

math.AG↗

arXiv2025

On universal deformation rings and stable homogeneous tubes

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

math.RT↗

arXiv2025

Valuations on Superrings

A valuation theory for superrings is developed, extending classical constructions from commutative algebra to the $\mathbb Z_2$-graded and supercommutative setting. We define valuations on superrings, investigate their fundamental properties, and explore the construction of Zariski-Riemann superspaces.

math.RA↗

arXiv2025

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

Let $\Bbbk$ be an algebraically closed field and $\Lambda$ a generalized Brauer tree algebra over $\Bbbk$. We compute the universal deformation rings of the periodic string modules over $\Lambda$. Moreover, for a specific class of generalized Brauer tree algebras $\Lambda(n,\overline{m})$, we classify the universal deformation rings of the modules lying in $\Omega$-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

El Teorema de Gelfand Naimark desde una perspectiva Categ\'orica The Gelfand--Naimark Theorem from a Categorical Perspective

Este art\'iculo presenta como resultado principal la equivalencia entre, las categor\'ias de espacios topol\'ogicos Hausdorff-Compactos y la categor\'ia de las $C^*-$\'algebras conmutativas con unidad, producto de la ``traducci\'on'' en este lenguaje del teorema de Gelfand--Naimark presentado en 1943. Haremos un recorrido sobre las principales ideas del an\'alisis y el \'algebra, conjugadas con \'exito, en el estudio de la teor\'ia de \'Algebras de Banach. As\'i mismo estableceremos, a forma de conclusi\'on, diversas aplicaciones que resultan naturalmente posibles a la luz de la ``analog\'ia y generalizaci\'on'' que nos permiten la teor\'ia de categor\'ias. Palabras claves: $C^*$-algebras, Categor\'ias, Espacios Topol\'ogicos, Teorema de Gelfand-Naimark, Teor\'ia de Representaciones. The goal of this paper is to prove the categorical equivalence between the category of Hausdorff-Compact topological spaces and the category of Unital Commutative $C^*$-algebras. This equivalence can be interpreted as a way of rewriting the well known Gelfand-Naimark Theorem in a categorical language. We will present the basic concepts in the theory of Banach Algebras as a successful link between Analysis and Algebra. Likewise, we will show some applications due to this new perspective, highlighting the categorical connection through proofs of typical problems that don't have an easy solution in $C^*-$algebra. Keywords: Category Theory, $C^*$-algebras, Gelfand-Naimark Theorem, Topological Spaces, Representation Theory.

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

A Note on Unique Factorization in Superrings

In the realm of supercommutative superrings, this article investigates the unique factorization of elements. We build upon recent findings by Naser et. al. concerning similar results in noncommutative symmetric rings with zerodivisors, delving deeper into the ramifications. Strikingly, we demonstrate that any unique factorization superdomain necessarily takes the form of a superfield, further characterizing them as Artinian superrings with Krull superdimension 0 | d. Furthermore, we discover that a straightforward analogue of the well-known Auslander-Buchsbaum Theorem does not hold true in the supercommutative setting.

math.RA↗

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

Cochain complexes over a functor

In this paper we propose unifying the categories of cochain complexes $\text{Ch}(\mathcal{C})$ and modules $\widehat{A}\text{-mod}$ over a repetitive algebra $\widehat{A}$. Motivated by their striking similarities and importance, we introduce a novel category encompassing both. Our analysis explores key properties of this unified category, highlighting its parallels and divergences from the original structures. We study whether it preserves crucial aspects like limits, colimits, products, coproducts, and abelianity. Besides, we establish a family of projective and injective indecomposable objects within this framework. Moving beyond theoretical foundations, we examine the influence and interaction over these novel categories of the category of endofunctors and its monoidal structure. Finally, we explore the implications of our constructions over representation theory of algebras and algebraic geometry.

math.RT↗

arXiv2023

UMP Monomial Algebras: Combinatorial and Homological Consequences

In this paper, we apply the techniques developed in [5] to present several consequences of studying UMP algebras and the ramifications graph of a monomial bound quiver algebra. Specifically, we prove that every weakly connected component of the ramifications graph of a UMP monomial algebra is unilaterally connected. Furthermore, using the main result characterizing UMP algebras in the monomial context, we prove that the class of UMP algebras is equivalent to the class of special multiserial algebras when the algebra is a quadratic monomial algebra. Based on this equivalence and the classification of Chen-Shen-Zhou on Gorenstein projective modules in [6], we extend their results to the class of monomial special multiserial UMP algebras, where we use the analysis of homological properties on quadratic monomial algebras given by these authors.

math.RT↗

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

Dedekind Superrings and Related Concepts

This article investigates the properties of Dedekind superrings, invertible supermodules and projective supermodules within the $\mathbb{Z}_2$-graded framework. Rather than treating these entities as specialized instances of general noncommutative ring theory, we develop them intrinsically within the category of supercommutative superrings. We examine the structural parallels to the classical commutative framework and, more importantly, characterize the fundamental discrepancies that emerge in the $\mathbb{Z}_2$-graded setting. In particular, we show that many hallmark equivalences of classical Dedekind domains-including those involving integral closedness and the coincidence of principal and unique factorization domains-fail to persist in the presence of an odd part.

math.RA↗

arXiv2023

About UMP algebras and a special classification case

The class of UMP algebras arises in several classification problems in the context of derived categories of finite-dimensional algebras. In this paper we define the class of UMP algebras and develop algebraic combinatorics tools in order to present a characterization of this class of algebras which are locally monomial (see Definition 4.1) and special multiserial algebras. Among other things, we describe the ramifications graph of symmetric special biserial algebras and we classify which of them are UMP algebras in terms of their bound quivers and their associated Brauer graphs.

math.RT↗

arXiv2023

A note on deformations of finite dimensional modules over $\mathbb{k}$-algebras

Let $\mathbb{k}$ be a field, and let $Λ$ be a (not necessarily finite dimensional) $\mathbb{k}$-algebra. Let $V$ be a left $Λ$-module such that is finite dimensional over $\mathbb{k}$. Assume further that $V$ has a weak universal deformation ring $R^w(Λ,V)$, which is a complete Noetherian commutative local $\mathbb{k}$-algebra with residue field $\mathbb{k}$. We prove in this note that under certain conditions on the $Λ$-module $V$, that if $R^w(Λ,V)$ is a quotient of $\mathbb{k}[\![t]\!]$, then $R^w(Λ,V)$ is either isomorphic to $\mathbb{k}$, or $\mathbb{k}[\![t]\!]$, or to $\mathbb{k}[\![t]\!]/(t^N)$ for some integer $N\geq 2$.

math.RT↗

arXiv2021

A note on the definability of genus for Zariski geometries

In this work we propose a notion of genus in the context of Zariski geometries and we obtain natural generalizations of the Riemann--Hurwitz Theorem and the Hurwitz Theorem in the context of very ample Zariski geometries. As a corollary, we show that such notion of genus cannot be first-order definable in the full language of a Zariski geometry.

math.LO↗

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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