arXiv subjects

Francesca Fedele

Publications and source records attributed to Francesca Fedele.

14 recordsLinked to original sources

arXiv2025

Lattices of pretorsion classes

Since their introduction, torsion theories have played a key role in the study of abelian and pointed categories. In representation theory, torsion theories and lattices of torsion classes of mod$ A$, for $A$ a finite-dimensional algebra, have been widely studied. The more recent definition of pretorsion theories, that can be given for any category, has expanded the theory, giving many more instances of ``non-pointed torsion theories'' in unexpected settings. In this work, we introduce and study the lattice $\mathcal{L}_t(A)$ of pretorsion classes of mod$ A$. These lattices are in close connection with the lattices tors$ A$ of torsion classes of mod$ A$. We fully describe the completely join-irreducible elements of $\mathcal{L}_t(A)$. Moreover, we characterise and give a full classification of when $\mathcal{L}_t(A)$ is distributive and further describe when it can be identified with the \emph{distributive closure} of tors$ A$. Finally, we show how the lattices of pretorsion classes, together with their duals, can be used to build pretorsion theories in mod$ A$.

math.RT↗

arXiv2024

Presentations of the braid group of the complex reflection group $G(d,d,n)$

We show that the braid group associated to the complex reflection group $G(d,d,n)$ is an index $d$ subgroup of the braid group of the orbifold quotient of the complex numbers by a cyclic group of order $d$. We also give a compatible presentation of $G(d,d,n)$ and its braid group for each tagged triangulation of the disk with $n$ marked points on its boundary and an interior marked point (interpreted as a cone point of degree $d$) in such a way that the presentations of Brou\'{e}-Malle-Rouquier correspond to a special tagged triangulation.

math.RT↗

arXiv2024

The index in $d$-exact categories

Starting from its original definition in module categories with respect to projective modules, the index has played an important role in various aspects of homological algebra, categorification of cluster algebras and $K$-theory. In the last few years, the notion of index has been generalised to several different contexts in (higher) homological algebra, typically with respect to a (higher) cluster-tilting subcategory $\mathcal{X}$ of the relevant ambient category $\mathcal{C}$. The recent tools of extriangulated and higher-exangulated categories have permitted some conditions on the subcategory $\mathcal{X}$ to be relaxed. In this paper, we introduce the index with respect to a generating, contravariantly finite subcategory of a $d$-exact category that has $d$-kernels. We show that our index has the important property of being additive on $d$-exact sequences up to an error term.

math.RT↗

arXiv2024

Super Caldero--Chapoton map for type $A$

One can explicitly compute the generators of a surface cluster algebra either combinatorially, through dimer covers of snake graphs, or homologically, through the CC-map applied to indecomposable modules over the appropriate algebra. Recent work by Musiker, Ovenhouse and Zhang used Penner and Zeitlin's decorated super Teichm{\"u}ller theory to define a super version of the cluster algebra of type $A$ and gave a combinatorial formula to compute the even generators. We extend this theory by giving a homological way of explicitly computing these generators by defining a super CC-map for type $A$.

math.RT↗

arXiv2024

The index with respect to a contravariantly finite subcategory

Cluster algebras are categorified by cluster categories, and $g$-vectors are categorified by the classic index with respect to cluster tilting subcategories. However, the recently introduced completed discrete cluster categories of Dynkin type $\mathbb{A}$ have a very limited supply of cluster tilting subcategories, so we define the index with respect to additive, contravariantly finite subcategories of which there are many more. This permits us to extend several strong results from the classic theory to completed discrete cluster categories of Dynkin type $\mathbb{A}$. Notably, the index with respect to the subcategory generated by a fan triangulation distinguishes between rigid objects. We also prove that our index is additive on triangles up to an error term. This extends the key property which permits the classic index to be used in the categorification of cluster algebras.

math.RT↗

arXiv2023

Building pretorsion theories from torsion theories

Torsion theories play an important role in abelian categories and they have been widely studied in the last sixty years. In recent years, with the introduction of pretorsion theories, the definition has been extended to general (non-pointed) categories. Many examples have been investigated in several different contexts, such as topological spaces and topological groups, internal preorders, preordered groups, toposes, V-groups, crossed modules, etc. In this paper, we show that pretorsion theories naturally appear also in the "classical" framework, namely in abelian categories. We propose two ways of obtaining pretorsion theories starting from torsion theories. The first one uses "comparable" torsion theories, while the second one extends a torsion theory with a Serre subcategory. We also give a universal way of obtaining a torsion theory from a given pretorsion theory in additive categories. We conclude by providing several applications in module categories, internal groupoids, recollements and representation theory.

math.CT↗

arXiv2022

Universal localizations of $d$-homological pairs

Let $k$ be an algebraically closed field and $\Phi$ a finite dimensional $k$-algebra. The universal localization $\Phi\rightarrow \Phi_\mathcal{S}$ of $\Phi$ with respect to a set of morphisms between finitely generated projective $\Phi$-modules $\mathcal{S}$ always exists. Moreover, when $\Phi$ is hereditary, Krause and \v{S}\v{t}ov\'i\v{c}ek proved that the universal localizations of $\Phi$ are in bijective correspondence with various natural structures. Taking inspiration from an alternative definition of universal localizations involving a triangulated subcategory of $\mathcal{D}^{\text{perf}}(\Phi)$, we introduce a higher analogue of universal localizations. That is, fixing a positive integer $d$, we define universal localizations of $d$-homological pairs $(\Phi,\mathcal{F})$ with respect to suitable wide subcategories $\mathcal{U}$ of $\mathcal{D}^b(\text{mod}\Phi)$. When gldim$\Phi\leq d$, we show that the result by Krause and \v{S}\v{t}ov\'i\v{c}ek has a (partial) higher analogue and that such universal localizations exist with respect to any choice of $\mathcal{U}$ with the required properties. Moreover, we show that in this setup, the base case of our definition and the definition of classic universal localization coincide.

math.RT↗

arXiv2020

Negative cluster categories from simple minded collection quadruples

Fomin and Zelevinsky's definition of cluster algebras laid the foundation for cluster theory. The various categorifications and generalisations of the original definition led to Iyama and Yoshino's generalised cluster categories $\mathcal{T}/\mathcal{T}^{fd}$ coming from positive-Calabi-Yau triples $(\mathcal{T}, \mathcal{T}^{fd},\mathcal{M})$. Jin later defined simple minded collection quadruples $(\mathcal{T}, \mathcal{T}^{p},\mathbb{S},\mathcal{S})$, where the special case $\mathbb{S}=\Sigma^{-d}$ is the analogue of Iyama and Yang's triples: negative-Calabi-Yau triples. In this paper, we further study the quotient categories $\mathcal{T}/\mathcal{T}^p$ coming from simple minded collection quadruples. Our main result uses limits and colimits to describe Hom-spaces over $\mathcal{T}/\mathcal{T}^p$ in relation to the easier to understand Hom-spaces over $\mathcal{T}$. Moreover, we apply our theorem to give a different proof of a result by Jin: if we have a negative-Calabi-Yau triple, then $\mathcal{T}/\mathcal{T}^p$ is a negative cluster category.

math.RT↗

arXiv2020

Grothendieck groups of triangulated categories via cluster tilting subcategories

Let $k$ be a field and $\mathcal{C}$ a $k$-linear, Hom-finite triangulated category with split idempotents. In this paper, we show that under suitable circumstances, the Grothendieck group of $\mathcal{C}$, denoted $K_0(\mathcal{C})$, can be expressed as a quotient of the split Grothendieck group of a higher-cluster tilting subcategory of $\mathcal{C}$. Assume that $n\geq 2$ is an even integer, $\mathcal{C}$ is $n$-Calabi Yau and has an $n$-cluster tilting subcategory $\mathcal{T}$. Then, for every indecomposable $M$ in $\mathcal{T}$, there is an Auslander-Reiten $(n+2)$-angle in $\mathcal{T}$ of the form $M\rightarrow T_{n-1}\rightarrow\dots\rightarrow T_0\rightarrow M$ and \begin{align*} K_0(\mathcal{C})\cong K_0^{sp}(\mathcal{T})\big/\big \langle \sum_{i=0}^{n-1}(-1)^i[T_i]\mid M\in\mathcal{T} \text{ indecomposable } \big\rangle. \end{align*} Assume now that $d$ is a positive integer and $\mathcal{C}$ has a $d$-cluster tilting subcategory $\mathcal{S}$ closed under $d$-suspension. Then $\mathcal{S}$ is a so called $(d+2)$-angulated category whose Grothendieck group $K_0(\mathcal{S})$ can be defined as a certain quotient of $K_0^{sp}(\mathcal{S})$. We will show \begin{align*} K_0(\mathcal{C})\cong K_0(\mathcal{S}). \end{align*} Moreover, assume that $n=2d$, that all the above assumptions hold, and that $\mathcal{T}\subseteq \mathcal{S}$. Then our results can be combined to express $K_0(\mathcal{S})$ as a quotient of $K_0^{sp}(\mathcal{T})$.

math.RT↗

arXiv2020

Properties of triangulated and quotient categories arising from $n$-Calabi-Yau triples

The original definition of cluster algebras by Fomin and Zelevinsky has been categorified and generalised in several ways over the course of the past 20 years, giving rise to cluster theory. This study lead to Iyama and Yang's generalised cluster categories $\mathcal{T}/\mathcal{T}^{fd}$ coming from $n$-Calabi-Yau triples $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$. In this paper, we use some classic tools of homological algebra to give a deeper understanding of such categories $\mathcal{T}/\mathcal{T}^{fd}$. Let $k$ be a field, $n\geq 3$ an integer and $\mathcal{T}$ a $k$-linear triangulated category with a triangulated subcategory $\mathcal{T}^{fd}$ and a subcategory $\mathcal{M}=\text{add}(M)$ such that $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$ is an $n$-Calabi-Yau triple. In this paper, we prove some properties of the triangulated categories $\mathcal{T}$ and $\mathcal{T}/\mathcal{T}^{fd}$. Our first result gives a relation between the Hom-spaces in these categories, using limits and colimits. Our second result is a Gap Theorem in $\mathcal{T}$, showing when the truncation triangles split. Moreover, we apply our two theorems to present an alternative proof to a result by Guo, originally stated in a more specific setup of dg $k$-algebras $A$ and subcategories of the derived category of dg $A$-modules. This proves that $\mathcal{T}/\mathcal{T}^{fd}$ is Hom-finite and $(n-1)$-Calabi-Yau, its object $M$ is $(n-1)$-cluster tilting and the endomorphism algebras of $M$ over $\mathcal{T}$ and over $\mathcal{T}/\mathcal{T}^{fd}$ are isomorphic. Note that these properties make $\mathcal{T}/\mathcal{T}^{fd}$ a generalisation of the cluster category.

math.RT↗

arXiv2020

$d$-Auslander-Reiten sequences in subcategories

Let $Φ$ be a finite dimensional algebra over a field $k$. Kleiner described the Auslander-Reiten sequences in a precovering extension closed subcategory $\mathcal{X}\subseteq$ mod $Φ$. If $X\in\mathcal{X}$ is an indecomposable such that Ext$_Φ^1(X,\mathcal{X})\neq 0$ and $ζX$ is the unique indecomposable direct summand of the $\mathcal{X}$-cover $g:Y\rightarrow D$Tr$X$ such that Ext$_Φ^1(X,ζX)\neq 0$, then there is an Auslander-Reiten sequence in $\mathcal{X}$ of the form \begin{align*} ε: 0\rightarrow ζX\rightarrow X'\rightarrow X\rightarrow 0. \end{align*} Moreover, when End$_Φ(X)$ modulo the morphisms factoring through a projective is a division ring, Kleiner proved that each non-split short exact sequence of the form \begin{align*} δ: 0\rightarrow Y\rightarrow Y'\xrightarrowη X\rightarrow 0 \end{align*} is such that $η$ is right almost split in $\mathcal{X}$, and the pushout of $δ$ along $g$ gives an Auslander-Reiten sequence in mod $Φ$ ending at $X$. In this paper, we give higher dimensional generalisations of this. Let $d\geq 1$ be an integer. A $d$-cluster tilting subcategory $\mathcal{F}\subseteq$ mod $Φ$ plays the role of a higher mod $Φ$. Such an $\mathcal{F}$ is a $d$-abelian category, where kernels and cokernels are replaced by complexes of $d$ objects and short exact sequences by complexes of $d+2$ objects. We give higher versions of the above results for an additive "$d$-extension closed" subcategory $\mathcal{X}$ of $\mathcal{F}$.

math.RT↗

arXiv2018

Auslander-Reiten $(d+2)$-angles in subcategories and a $(d+2)$-angulated generalisation of a theorem by Brüning

Let $Φ$ be a finite dimensional algebra over an algebraically closed field $k$ and assume gldim$\,Φ\leq d$, for some fixed positive integer $d$. For $d=1$, Brüning proved that there is a bijection between the wide subcategories of the abelian category mod$\,Φ$ and those of the triangulated category $\mathcal{D}^b(\text{mod}Φ)$. Moreover, for a suitable triangulated category $\mathcal{M}$, Jørgensen gave a description of Auslander-Reiten triangles in the extension closed subcategories of $\mathcal{M}$. In this paper, we generalise these results for $d$-abelian and $(d+2)$-angulated categories, where kernels and cokernels are replaced by complexes of $d+1$ objects and triangles are replaced by complexes of $d+2$ objects. The categories are obtained as follows: if $\mathcal{F}\subseteq \text{mod} Φ$ is a $d$-cluster tilting subcategory, consider $\overline{\mathcal{F}}:=\text{add} \{Σ^{id}\mathcal{F}\mid i\in\mathbb{Z} \}\subseteq \mathcal{D}^b(\text{mod}Φ)$. Then $\mathcal{F}$ is $d$-abelian and plays the role of a higher mod$\,Φ$ having for higher derived category the $(d+2)$-angulated category $\overline{\mathcal{F}}$.

math.RT↗

arXiv2017

Almost split morphisms in subcategories of triangulated categories

For a suitable triangulated category $\mathcal{T}$ with a Serre functor $S$ and a full precovering subcategory $\mathcal{C}$ closed under summands and extensions, an indecomposable object $C$ in $\mathcal{C}$ is called Ext-projective if Ext$^1(C,\mathcal{C})=0$. Then there is no Auslander-Reiten triangle in $\mathcal{C}$ with end term $C$. In this paper, we show that if, for such an object $C$, there is a minimal right almost split morphism $\beta:B\rightarrow C$ in $\mathcal{C}$, then $C$ appears in something very similar to an Auslander-Reiten triangle in $\mathcal{C}$: an essentially unique triangle in $\mathcal{T}$ of the form \begin{align*} \Delta= X\xrightarrow{\xi} B\xrightarrow{\beta} C\rightarrow \Sigma X, \end{align*} where $X$ is an indecomposable not in $\mathcal{C}$ and $\xi$ is a $\mathcal{C}$-envelope of $X$. Moreover, under some extra assumptions, we show that removing $C$ from $\mathcal{C}$ and replacing it with $X$ produces a new subcategory of $\mathcal{T}$ closed under extensions. We prove that this process coincides with the classic mutation of $\mathcal{C}$ with respect to the rigid subcategory of $\mathcal{C}$ generated by all the indecomposable Ext-projectives in $\mathcal{C}$ apart from $C$. When $\mathcal{T}$ is the cluster category of Dynkin type $A_n$ and $\mathcal{C}$ has the above properties, we give a full description of the triangles in $\mathcal{T}$ of the form $\Delta$ and show under which circumstances replacing $C$ by $X$ gives a new extension closed subcategory.

math.RT↗

arXiv2017

Distributions-oriented wind forecast verification by a hidden Markov model for multivariate circular-linear data

Winds from the North-West quadrant and lack of precipitation are known to lead to an increase of PM10 concentrations over a residential neighborhood in the city of Taranto (Italy). In 2012 the local government prescribed a reduction of industrial emissions by 10% every time such meteorological conditions are forecasted 72 hours in advance. Wind forecasting is addressed using the Weather Research and Forecasting (WRF) atmospheric simulation system by the Regional Environmental Protection Agency. In the context of distributions-oriented forecast verification, we propose a comprehensive model-based inferential approach to investigate the ability of the WRF system to forecast the local wind speed and direction allowing different performances for unknown weather regimes. Ground-observed and WRF-forecasted wind speed and direction at a relevant location are jointly modeled as a 4-dimensional time series with an unknown finite number of states characterized by homogeneous distributional behavior. The proposed model relies on a mixture of joint projected and skew normal distributions with time-dependent states, where the temporal evolution of the state membership follows a first order Markov process. Parameter estimates, including the number of states, are obtained by a Bayesian MCMC-based method. Results provide useful insights on the performance of WRF forecasts in relation to different combinations of wind speed and direction.

stat.AP↗