arXiv subjects

Raquel Coelho Simoes

Publications and source records attributed to Raquel Coelho Simoes.

12 recordsLinked to original sources

arXiv2026

Simple tilts of length hearts and simple-minded mutation

We characterise when a simple Happel-Reiten-Smalo tilt of a length heart is again a length heart in terms of approximation theory and the existence of a stability condition with a phase gap. We apply simple-minded reduction to provide a sufficient condition for infinite iterability of simple-minded mutation/simple tilting. We use simple-minded mutation pairs to provide a common framework to show that mutation of simple-minded collections (resp. $w$-simple-minded systems, for $w \geq 1$) gives simple-minded collections (resp. $w$-simple-minded systems) under mild conditions, in the process providing a unified proof of results of Alex Dugas and Peter Jorgensen. Finally, we show that under mild conditions, mutation of simple-minded collections is compatible with mutation of $w$-simple-minded systems via a singularity category construction due to Haibo Jin.

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arXiv2024

Maximal almost rigid modules over gentle algebras

We study maximal almost rigid modules over a gentle algebra $A$. We prove that the number of indecomposable direct summands of every maximal almost rigid $A$-module is equal to the sum of the number of vertices and the number of arrows of the Gabriel quiver of $A$. Moreover, the algebra $A$, considered as an $A$-module, can be completed to a maximal almost rigid module in a unique way. Gentle algebras are precisely the tiling algebras of surfaces with marked points. We show that the (permissible) triangulations of the surface of $A$ are in bijection with the maximal almost rigid $A$-modules. Furthermore, we study the endomorphism algebra $C=\text{End}_A T$ of a maximal almost rigid module $T$. We construct a fully faithful functor $G\colon \text{mod}\,A\to \text{mod}\, \overline{A}$ into the module category of a bigger gentle algebra $\overline{A}$ and show that $G$ maps maximal almost rigid $A$-modules to tilting $\overline{A}$-modules. In particular, $C$ and $\overline{A}$ are derived equivalent and $C$ is gentle. After giving a geometric realization of the functor $G$, we obtain a tiling $G(\mathbf{T})$ of the surface of $\overline{A}$ as the image of the triangulation $\mathbf{T}$ corresponding to $T$. We then show that the tiling algebra of $G(\mathbf{T})$ is $C$. Moreover, the tiling algebra of $\mathbf{T}$ is obtained algebraically from $C$ as the tensor algebra with respect to the $C$-bimodule $\text{Ext}_C^2(DC,C)$, which also is fundamental in cluster-tilting theory.

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arXiv2024

A geometric model for the module category of a string algebra

In this paper, we give a geometric construction of string algebras and of their module categories. Our approach uses dissections of punctured Riemann surfaces with extra data at marked points, called labels. As an application, we give a classification of support tau-tilting modules in terms of arcs in such a tiled surface. In the case when the string algebra is gentle, we recover the classification given arXiv:2004.11136.

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arXiv2021

Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions

Let $Q$ be an acyclic quiver and $w \geq 1$ be an integer. Let $\mathsf{C}_{-w} (\mathbf{k} Q)$ be the $(-w)$-cluster category of $\mathbf{k} Q$. We show that there is a bijection between simple-minded collections in $\mathsf{D}^b (\mathbf{k} Q)$ lying in a fundamental domain of $\mathsf{C}_{-w} (\mathbf{k} Q)$ and $w$-simple-minded systems in $\mathsf{C}_{-w} (\mathbf{k} Q)$. This generalises the same result of Iyama-Jin in the case that $Q$ is Dynkin. A key step in our proof is the observation that the heart $\mathsf{H}$ of a bounded t-structure in a Hom-finite, Krull-Schmidt, $\mathbf{k}$-linear saturated triangulated category $\mathsf{D}$ is functorially finite in $\mathsf{D}$ if and only if $\mathsf{H}$ has enough injectives and enough projectives. We then establish a bijection between $w$-simple-minded systems in $\mathsf{C}_{-w} (\mathbf{k} Q)$ and positive $w$-noncrossing partitions of the corresponding Weyl group $W_Q$.

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arXiv2018

Simple-minded systems and reduction for negative Calabi-Yau triangulated categories

We develop the basic properties of $w$-simple-minded systems in $(-w)$-Calabi-Yau triangulated categories for $w \geq 1$. The main result is a reduction technique for negative Calabi-Yau triangulated categories. We show that the theory of simple-minded systems exhibits striking parallels with that of cluster-tilting objects. Our construction provides an inductive technique for constructing simple-minded systems.

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arXiv2018

A geometric model for the module category of a gentle algebra

In this article, gentle algebras are realised as tiling algebras, which are associated to partial triangulations of unpunctured surfaces with marked points on the boundary. This notion of tiling algebras generalise the notion of Jacobian algebras of triangulations of surfaces and the notion of surface algebras. We use this description to give a geometric model of the module category of any gentle algebra.

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arXiv2016

Endomorphism algebras for a class of negative Calabi-Yau categories

We consider an orbit category of the bounded derived category of a path algebra of type A_n which can be viewed as a -(m+1)-cluster category, for m >= 1. In particular, we give a characterisation of those maximal m-rigid objects whose endomorphism algebras are connected, and then use it to explicitly study these algebras. Specifically, we give a full description of them in terms of quivers and relations, and relate them with (higher) cluster-tilted algebras of type A. As a by-product, we introduce a larger class of algebras, called 'tiling algebras'.

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arXiv2015

Mutations of simple-minded systems in Calabi-Yau categories generated by a spherical object

In this article, we give a definition and a classification of 'higher' simple-minded systems in triangulated categories generated by spherical objects with negative Calabi-Yau dimension. We also study mutations of this class of objects and that of 'higher' Hom-configurations and Riedtmann configurations. This gives an explicit analogue of the nice mutation theory exhibited in cluster-tilting theory.

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arXiv2015

Torsion pairs in a triangulated category generated by a spherical object

We extend Ng's characterisation of torsion pairs in the 2-Calabi-Yau triangulated category generated by a 2-spherical object to the characterisation of torsion pairs in the w-Calabi-Yau triangulated category, $T_w$, generated by a w-spherical object for any integer w. Inspired by the combinatorics of $T_w$ for w < 0, we also characterise the torsion pairs in a certain w-Calabi-Yau orbit category of the bounded derived category of the path algebra of Dynkin type A.

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arXiv2013

Hom-configurations in triangulated categories generated by spherical objects

Hom- and Riedtmann configurations were studied in the context of stable module categories of selfinjective algebras and a certain orbit category C of the bounded derived category of a Dynkin quiver, which is highly reminiscent of the cluster category. The category C is (-1)-Calabi-Yau. Holm and Jorgensen introduced a family of triangulated categories generated by $w$-spherical objects. When $w \geq 2$, these may be regarded as higher cluster categories of type A infinity. When $w \leq -1$, they are higher analogues of the orbit category C. In this paper, we classify the (higher) Hom- and Riedtmann configurations for these categories, and link them with noncrossing partitions in the case $w = -1$. Along the way, we obtain a new geometric model for the higher versions of the orbit category C.

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arXiv2011

Maximal rigid objects as noncrossing bipartite graphs

Let Q be a Dynkin quiver of type A. The bounded derived category of the path algebra of Q has an autoequivalence given by the composition of the Auslander-Reiten translate and the square of the shift functor. We classify the maximal rigid objects in the corresponding orbit category C(Q), in terms of bipartite noncrossing graphs (with loops) in a circle. We also describe the endomorphism algebras of the maximal rigid objects, and we prove that a certain class of these algebras are iterated tilted algebras of type A.

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arXiv2010

Hom-configurations and noncrossing partitions

Let Q be a Dynkin quiver. The bounded derived category of the path algebra of Q has an autoequivalence given by the composition of the Auslander-Reiten translate and the square of the shift functor. We study maximal Hom-free sets in the corresponding orbit category C(Q). We prove that these sets are in bijection with periodic combinatorial configurations, as introduced by Riedtmann, certain Hom<=0-configurations, studied by Buan, Reiten and Thomas, and noncrossing partitions of the Coxeter group associated to Q which are not contained in any proper standard parabolic subgroup. Note that Reading has proved that these noncrossing partitions are in bijection with positive clusters in the associated cluster algebra. Finally, we give a definition of mutations of maximal Hom-free sets in C(Q) and prove that the graph of these mutations is connected.

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