arXiv subjects

Pedro Rizzo

Publications and source records attributed to Pedro Rizzo.

22 records · Page 2Linked to original sources

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↗

arXiv2016

Level-$δ$ limit linear series

We introduce the notion of level-$δ$ limit linear series, which describe limits of linear series along families of smooth curves degenerating to a singular curve $X$. We treat here only the simplest case where $X$ is the union of two smooth components meeting transversely at a point $P$. The integer $δ$ stands for the singularity degree of the total space of the degeneration at $P$. If the total space is regular, we get level-1 limit linear series, which are precisely those introduced by Osserman in 2006. We construct a projective moduli space $G^r_{d,δ}(X)$ parameterizing level-$δ$ limit linear series of rank $r$ and degree $d$ on $X$, and show that it is a new compactification, for each $δ$, of the moduli space of Osserman exact limit linear series, an open subscheme $G^{r,*}_{d,1}(X)$ of the space $G^r_{d,1}(X)$ already constructed by Osserman. Finally, we generalize work by Esteves and Osserman by associating to each exact level-$δ$ limit linear series $\mathfrak g$ on $X$ a closed subscheme $\mathbb P(\mathfrak g)\subseteq X^{(d)}$ of the $d$th symmetric product of $X$, and showing that $\mathbb P(\mathfrak g)$ is the limit of the spaces of divisors associated to linear series on smooth curves degenerating to $\mathfrak g$ on $X$, if such degenerations exist. In particular, we describe completely limits of divisors along degenerations to such a curve $X$.

math.AG↗