arXiv subjects

Charles Paquette

Publications and source records attributed to Charles Paquette.

35 records · Page 2Linked to original sources

arXiv2021

Completions of discrete cluster categories of type $\mathbb{A}$

We complete the discrete cluster categories of type $\mathbb{A}$ as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then taking a certain Verdier quotient. The resulting category is a Hom-finite Krull-Schmidt triangulated category containing the discrete cluster category as a full subcategory. The objects and Hom-spaces in this new category can be described geometrically, even though the category is not $2$-Calabi-Yau and Ext-spaces are not always symmetric. We describe all cluster-tilting subcategories. Given such a subcategory, we define a cluster character that takes values in a ring with infinitely many indeterminates. Our cluster character is new in that it takes into account infinite dimensional sub-representations of infinite dimensional ones. We show that it satisfies the multiplication formula and also the exchange formula, provided that the objects being exchanged satisfy some local Calabi-Yau conditions.

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arXiv2019

Idempotent reduction for the finitistic dimension conjecture

In this note, we prove that if $Λ$ is an Artin algebra with a simple module $S$ of finite projective dimension, then the finiteness of the finitistic dimension of $Λ$ implies that of $(1-e)Λ(1-e)$ where $e$ is the primitive idempotent supporting $S$. We derive some consequences of this. In particular, we recover a result of Green-Solberg-Psaroudakis: if $Λ$ is the quotient of a path algebra by an admissible ideal $I$ whose defining relations do not involve a certain arrow $α$, then the finitistic dimension of $Λ$ is finite if and only if the finitistic dimension of $Λ/ΛαΛ$ is finite.

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arXiv2018

Group actions on cluster algebras and cluster categories

We introduce admissible group actions on cluster algebras, cluster categories and quivers with potential and study the resulting orbit spaces. The orbit space of the cluster algebra has the structure of a generalized cluster algebra. This generalized cluster structure is different from those introduced by Chekhov-Shapiro and Lam-Pylyavskyy. For group actions on cluster algebras from surfaces, we describe the generalized cluster structure of the orbit space in terms of a triangulated orbifold. In this case, we give a complete list of exchange polynomials, and we classify the algebras of rank 1 and 2. We also show that every admissible group action on a cluster category induces a precovering from the cluster category to the cluster category of orbits. Moreover this precovering is dense if the categories are of finite type.

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arXiv2017

Generators versus projective generators in abelian categories

Let $\mathcal{A}$ be an essentially small abelian category. We prove that if $\mathcal{A}$ admits a generator $M$ with ${\rm End}_{\mathcal{A}}(M)$ right artinian, then $\mathcal{A}$ admits a projective generator. If $\mathcal{A}$ is further assumed to be Grothendieck, then this implies that $\mathcal{A}$ is equivalent to a module category. When $\mathcal{A}$ is Hom-finite over a field $k$, the existence of a generator is the same as the existence of a projective generator, and in case there is such a generator, $\mathcal{A}$ has to be equivalent to the category of finite dimensional right modules over a finite dimensional $k$-algebra. We also show that when $\mathcal{A}$ is a length category, then there is a one-to-one correspondence between exact abelian extension closed subcategories of $\mathcal{A}$ and collections of Hom-orthogonal Schur objects in $\mathcal{A}$.

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arXiv2017

Homological behavior of idempotent subalgebras and Ext algebras

Let $A$ be a (left and right) Noetherian ring that is semiperfect. Let $e$ be an idempotent of $A$ and consider the ring $Γ:=(1-e)A(1-e)$ and the semi-simple right $A$-module $S_e : = eA/e{\rm rad}A$. In this paper, we investigate the relationship between the global dimensions of $A$ and $Γ$, by using the homological properties of $S_e$. More precisely, we consider the Yoneda ring $Y(e):={\rm Ext}^*_A(S_e,S_e)$ of $e$. We prove that if $Y(e)$ is artinian of finite global dimension, then $A$ has finite global dimension if and only if so is $Γ$. We also investigate the situation where both $A,Γ$ have finite global dimension. When $A$ is Koszul and finite dimensional, this implies that $Y(e)$ has finite global dimension. We end the paper with a reduction technique to compute the Cartan determiant of artin algebras. We prove that if $Y(e)$ has finite global dimension, then the Cartan determinants of $A$ and $Γ$ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture.

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arXiv2016

Isotropic Schur roots

In this paper, we study the isotropic Schur roots of an acyclic quiver $Q$ with $n$ vertices. We study the perpendicular category $\mathcal{A}(d)$ of a dimension vector $d$ and give a complete description of it when $d$ is an isotropic Schur $δ$. This is done by using exceptional sequences and by defining a subcategory $\mathcal{R}(Q,δ)$ attached to the pair $(Q,δ)$. The latter category is always equivalent to the category of representations of a connected acyclic quiver $Q_{\mathcal{R}}$ of tame type, having a unique isotropic Schur root, say $δ_{\mathcal{R}}$. The understanding of the simple objects in $\mathcal{A}(δ)$ allows us to get a finite set of generators for the ring of semi-invariants SI$(Q,δ)$ of $Q$ of dimension vector $δ$. The relations among these generators come from the representation theory of the category $\mathcal{R}(Q,δ)$ and from a beautiful description of the cone of dimension vectors of $\mathcal{A}(δ)$. Indeed, we show that SI$(Q,δ)$ is isomorphic to the ring of semi-invariants SI$(Q_{\mathcal{R}},δ_{\mathcal{R}})$ to which we adjoin variables. In particular, using a result of Skowroński and Weyman, the ring SI$(Q,δ)$ is a polynomial ring or a hypersurface. Finally, we provide an algorithm for finding all isotropic Schur roots of $Q$. This is done by an action of the braid group $B_{n-1}$ on some exceptional sequences. This action admits finitely many orbits, each such orbit corresponding to an isotropic Schur root of a tame full subquiver of $Q$.

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arXiv2016

Irreducible morphisms and locally finite dimensional representations

Let $\mathcal{A}$ be a Hom-finite additive Krull-Schmidt $k$-category where $k$ is an algebraically closed field. Let ${\rm mod} \mathcal{A}$ denote the category of locally finite dimensional $\mathcal{A}$-modules, that is, the category of covariant functors $\mathcal{A} \to {\rm mod} k$. We prove that an irreducible monomorphism in ${\rm mod} \mathcal{A}$ has a finitely generated cokernel, and that an irreducible epimorphism in ${\rm mod} \mathcal{A}$ has a finitely co-generated kernel. Using this, we get that an almost split sequence in ${\rm mod} \mathcal{A}$ has to start with a finitely co-presented module and end with a finitely presented one. Finally, we apply our results in the study of ${\rm rep}(Q)$, the category of locally finite dimensional representations of a strongly locally finite quiver. We describe all possible shapes of the Auslander-Reiten quiver of ${\rm rep}(Q)$.

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arXiv2015

Homological dimensions for co-rank one idempotent subalgebras

Let $k$ be an algebraically closed field and $A$ be a (left and right) Noetherian associative $k$-algebra. Assume further that $A$ is either positively graded or semiperfect (this includes the class of finite dimensional $k$-algebras, and $k$-algebras that are finitely generated modules over a Noetherian central Henselian ring). Let $e$ be a primitive idempotent of $A$, which we assume is of degree $0$ if $A$ is positively graded. We consider the idempotent subalgebra $Γ= (1-e)A(1-e)$ and $S_e$ the simple right $A$-module $S_e = eA/e{\rm rad}A$, where ${\rm rad}A$ is the Jacobson radical of $A$, or the graded Jacobson radical of $A$ if $A$ is positively graded. In this paper, we relate the homological dimensions of $A$ and $Γ$, using the homological properties of $S_e$. First, if $S_e$ has no self-extensions of any degree, then the global dimension of $A$ is finite if and only if that of $Γ$ is. On the other hand, if the global dimensions of both $A$ and $Γ$ are finite, then $S_e$ cannot have self-extensions of degree greater than one, provided $A/{\rm rad}A$ is finite dimensional.

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arXiv2015

Cluster categories of type $\mathbb{A}_\infty^\infty$ and triangulations of the infinite strip

We first study the (canonical) orbit category of the bounded derived category of finite dimensional representations of a quiver with no infinite path, and we pay more attention on the case where the quiver is of infinite Dynkin type. In particular, its Auslander-Reiten components are explicitly described. When the quiver is of type $\mathbb{A}_\infty$ or $\mathbb{A}_\infty^\infty$, we show that this orbit category is a cluster category, that is, its cluster-tilting subcategories form a cluster structure. When the quiver is of type $\mathbb{A}_\infty^\infty$, we shall give a geometrical description of the cluster structure of the cluster category by using triangulations of the infinite strip in the plane. In particular, we shall show that the cluster-tilting subcategories are precisely given by compact triangulations.

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arXiv2015

Semi-stable subcategories for Euclidean quivers

In this paper, we study the semi-stable subcategories of the category of representations of a Euclidean quiver, and the possible intersections of these subcategories. Contrary to the Dynkin case, we find out that the intersection of semi-stable subcategories may not be semi-stable. However, only a finite number of exceptions occur, and we give a description of these subcategories. Moreover, one can attach a simplicial fan in $\mathbb{Q}^n$ to any acyclic quiver $Q$, and this simplicial fan allows one to completely determine the canonical presentation of any element in $\mathbb{Z}^n$. This fan has a nice description in the Dynkin and Euclidean cases: it is described using an arrangement of convex codimension-one subsets of $\mathbb{Q}^n$, each such subset being indexed by a real Schur root or a set of quasi-simple objects. This fan also characterizes when two different stability conditions give rise to the same semi-stable subcategory.

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arXiv2015

Accumulation points of real Schur roots

Let $k$ be an algebraically closed field and $Q$ be an acyclic quiver with $n$ vertices. Consider the category ${\rm rep}(Q)$ of finite dimensional representations of $Q$ over $k$. The exceptional representations of $Q$, that is, the indecomposable objects of ${\rm rep}(Q)$ without self-extensions, correspond to the so-called real Schur roots of the usual root system attached to $Q$. These roots are special elements of the Grothendieck group $\mathbb{Z}^n$ of ${\rm rep}(Q)$. When we identify the dimension vectors of the representations (that is, the non-negative vectors of $\mathbb{Z}^n$) up to positive multiple, we see that the real Schur roots can accumulate in some directions of $\mathbb{R}^n \supset \mathbb{Z}^n$. This paper is devoted to the study of these accumulation points. After giving new properties of the canonical decomposition of dimension vectors, we show how to use this decomposition to describe the rational accumulation points. Finally, we study the irrational accumulation points and we give a complete description of them in case $Q$ is of weakly hyperbolic type.

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arXiv2012

On the Auslander-Reiten quiver of the representations of an infinite quiver

Let Q be a strongly locally finite quiver and denote by rep(Q) the category of locally finite dimensional representations of Q over some fixed field k. The main purpose of this paper is to get a better understanding of rep(Q) by means of its Auslander-Reiten quiver. To achieve this goal, we define a category C(Q) which is a full, abelian and Hom-finite subcategory of rep(Q) containing all the almost split sequences of rep(Q). We give a complete description of the Auslander-Reiten quiver of C(Q) by describing its connected components. Finally, we prove that these connected components are also connected components of the Auslander-Reiten quiver of rep(Q). We end the paper by giving a conjecture describing the Auslander-Reiten components of rep(Q) that cannot be obtained as Auslander-Reiten components of C(Q).

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arXiv2012

Standard components of a Krull-Schmidt category

We provide criteria for an Auslander-Reiten component having sections of a Krull-Schmidt category to be standard. Specializing to the category of finitely presented representations of a strongly locally finite quiver and its bounded derived category, we obtain many new types of standard Auslander-Reiten components. An application to the module category of a finite-dimensional algebra yields some interesting results.

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arXiv2012

Almost split sequences and approximations

Let A be an exact category, that is, an extension-closed full subcategory of an abelian category. Firstly, we give some necessary and sufficient conditions for A to have almost split sequences. Then, we study when an almost split sequence in A induces an almost split sequence in an exact subcategory C of A. In case A has almost split sequences and C is Hom-finite Krull-Schmidt, this provides a necessary and sufficient condition for C to have almost split sequences. Finally, we show two applications of these results.

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arXiv2011

Representation Theory of an Infinite Quiver

This paper deals with the representation theory of a locally finite quiver in which the number of paths between any two given vertices is finite. We first study some properties of the finitely presented or co-presented representations, and then construct in the category of locally finite dimensional representations some almost split sequences which start with a finitely co-presented representation and end with a finitely presented representation. Furthermore, we obtain a general description of the shapes of the Auslander-Reiten components of the category of finitely presented representations and prove that the number of regular Auslander-Reiten components is infinite if and only if the quiver is not of finite or infinite Dynkin type. In the infinite Dynkin case, we shall give a complete list of the indecomposable representations and an explicit description of the Auslander-Reiten components. Finally, we apply these results to study the Auslander-Reiten theory in the derived category of bounded complexes of finitely presented representations.

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arXiv2011

A non-existence theorem for almost split sequences

Let k be a field, Q a quiver with countably many vertices and I an ideal of kQ such that kQ/I has finite dimensional Hom-spaces. In this note, we prove that there is no almost split sequence ending at an indecomposable not finitely presented representation of the bound quiver (Q,I). We then get that an indecomposable representation M of (Q,I) is the ending term of an almost split sequence if and only if it is finitely presented and not projective. The dual results are also true.

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arXiv2011

A proof of the strong no loop conjecture

The strong no loop conjecture states that a simple module of finite projective dimension over an artin algebra has no non-zero self-extension. The main result of this paper establishes this well known conjecture for finite dimensional algebras over an algebraically closed field.

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