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Register a new userClassifying torsion classes of gentle algebras
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For a finite-dimensional gentle algebra, it is already known that the functorially finite torsion classes of its category of finite-dimensional modules can be classified using a combinatorial interpretation, called maximal non-crossing sets of strings, of the corresponding support $tau$-tilting module (or equivalently, two-term silting complexes). In the topological interpretation of gentle algebras via marked surfaces, such a set can be interpreted as a dissection (or partial triangulation), or equivalently, a lamination that does not contain a closed curve. We will refine this combinatorics, which gives us a classification of torsion classes in the category of finite length modules over a (possibly infinite-dimensional) gentle algebra. As a consequence, our result also unifies the functorially finite torsion class classification of finite-dimensional gentle algebras with certain classes of special biserial algebras - such as Brauer graph algebras.
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Let $A$ be a finite-dimensional gentle algebra over an algebraically closed field. We investigate the combinatorial properties of support $tau$-tilting graph of $A$. In particular, it is proved that the support $tau$-tilting graph of $A$ is connected and has the so-called reachable-in-face property. The property was conjectured by Fomin and Zelevinsky for exchange graphs of cluster algebras which was recently confirmed by Cao and Li.
For any gentle algebra $Lambda=KQ/langle Irangle$, following Kalck, we describe the quiver and the relations for its Cohen-Macaulay Auslander algebra $mathrm{Aus}(mathrm{Gproj}Lambda)$ explicitly, and obtain some properties, such as $Lambda$ is representation-finite if and only if $mathrm{Aus}(mathrm{Gproj}Lambda)$ is; if $Q$ has no loop and any indecomposable $Lambda$-module is uniquely determined by its dimension vector, then any indecomposable $mathrm{Aus}(mathrm{Gproj}Lambda)$-module is uniquely determined by its dimension vector.
Let $K$ be an algebraically closed field. Let $(Q,Sp,I)$ be a skewed-gentle triple, $(Q^{sg},I^{sg})$ and $(Q^g,I^{g})$ be its corresponding skewed-gentle pair and associated gentle pair respectively. It proves that the skewed-gentle algebra $KQ^{sg}/< I^{sg}>$ is singularity equivalent to $KQ/< I>$. Moreover, we use $(Q,Sp,I)$ to describe the singularity category of $KQ^g/< I^g>$. As a corollary, we get that $mathrm{gldim} KQ^{sg}/< I^{sg}><infty$ if and only if $mathrm{gldim} KQ/< I><infty$ if and only if $mathrm{gldim} KQ^{g}/< I^{g}><infty$.
Let $Lambda$ be a finite-dimensional associative algebra. The torsion classes of $mod, Lambda$ form a lattice under containment, denoted by $tors, Lambda$. In this paper, we characterize the cover relations in $tors, Lambda$ by certain indecomposable modules. We consider three applications: First, we show that the completely join-irreducible torsion classes (torsion classes which cover precisely one element) are in bijection with bricks. Second, we characterize faces of the canonical join complex of $tors, Lambda$ in terms of representation theory. Finally, we show that, in general, the algebra $Lambda$ is not characterized by its lattice $tors, Lambda$. In particular, we study the torsion theory of a quotient of the preprojective algebra of type $A_n$. We show that its torsion class lattice is isomorphic to the weak order on $A_n$.
Following [20], a desingularization of arbitrary quiver Grassmannians for finite dimensional Gorenstein projective modules of 1-Gorenstein gentle algebras is constructed in terms of quiver Grassmannians for their Cohen-Macaulay Auslander algebras.
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