Related papers: Torsion pairs and rigid objects in tubes

We give a complete classification of torsion pairs in the cluster category of Dynkin type A_n. Along the way we give a new combinatorial description of Ptolemy diagrams, an infinite version of which was introduced by Ng. This allows us to…

We give a geometric model for the non-$\tau$-rigid modules over acyclic path algebras of type $\widetilde{D}_n$. Similar models have been provided for module categories over path algebras of types $A_n, D_n,$ and $\widetilde{A}_n$ as well…

We give a geometric realization of module categories of type $\tilde{A}_n$. We work with oriented arcs to define a translation quiver isomorphic to the Auslander-Reiten quiver of the module category of type $\tilde{A}_n$. To get a…

We study the lattice $\mathbf{tors}(A)$ of torsion pairs in the category $\mathrm{mod}(A)$ of finitely generated modules over an artinian ring $A$. It was shown by the authors in previous work that $\mathbf{tors}(A)$ is isomorphic to a…

We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair $(\mathcal{A},\mathcal{B})$ in an exact category $\mathcal{C}$, $\mathcal{A}$ coincides…

Let $\mathcal{M}$ be an abelian model category (in the sense of Hovey). For a large class of quivers, we describe associated abelian model structures on categories of quiver representations with values in $\mathcal{M}$. This is based on…

Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed…

Let $\mathcal{C}$ be a small category. In this paper, we mainly study the category of modules $\mathfrak{M}\mbox{od-}\mathfrak{R}$ on ringed sites $(\mathbf{C},\mathfrak{R})$. We firstly reprove the Theorem A of the paper (M. Wu and F. Xu.…

We construct the skein category $\mathcal{S}$ of the annulus and show that it is equivalent to the affine Temperley-Lieb category of Graham and Lehrer. It leads to a skein theoretic description of the extended affine Temperley-Lieb…

We define torsion pairs for quasi-abelian categories and give several characterisations. We show that many of the torsion theoretic concepts translate from abelian categories to quasi-abelian categories. As an application, we generalise the…

We give a geometric interpretation of sheaf cohomology for higher degrees n in terms of torsors on the member of degree d=n-1 in hypercoverings of type r=n-2, endowed with an additional data, the so-called rigidification. This generalizes…

In this article, we give a geometric model for non-homogeneous tubes of the cluster category of the affine type $D$. This model is given in terms of homotopy classes of unoriented arcs in the twice punctured disk. In particular, we extend…

Initiated in work by Adachi, Iyama and Reiten, the area known as $\tau$-tilting theory plays a fundamental role in contemporary representation theory. In this paper we explore a higher-dimensional analogue of this theory, formulated with…

We give a complete classification of torsion pairs in repetitive cluster categories of type $A_n$, which were defined by Zhu as the orbit categories, via certain configurations of diagonals, called Ptolemy diagrams. As applications, we…

Given a commutative ring R (respectively a positively graded commutative ring $A=\ps_{j\geq 0}A_j$ which is finitely generated as an A_0-algebra), a bijection between the torsion classes of finite type in Mod R (respectively tensor torsion…

Let $A$ be the path algebra of a quiver of Dynkin type $\mathbb{A}_n$. The module category $\text{mod}\,A$ has a combinatorial model as the category of diagonals in a polygon $S$ with $n+1$ vertices. The recently introduced notion of almost…

We show that torsion pairs in Krull--Schmidt abelian categories induce an equivalence between the subcategory of torsion-free objects admitting universal extensions to the torsion subcategory, and a quotient of the ext-orthogonal complement…

For a given modular tensor category we study representations of the corresponding tube category whose isomorphism classes are modular invariant matrices. In particular, we provide a characterization of these representations in terms of the…

We introduce the notion of ST-pairs of triangulated subcategories, a prototypical example of which is the pair of the bound homotopy category and the bound derived category of a finite-dimensional algebra. For an ST-pair $(\C,\D)$, we…

We give an interpretation of the map $\pi^c$ defined by Reading, which is a map from the elements of a Coxeter group to the $c$-sortable elements, in terms of the representation theory of preprojective algebras. Moreover, we study a close…