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Register a new userOn support $tau$-tilting graphs of gentle 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.
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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.
We discuss tilting modules of affine quasi-hereditary algebras. We present an existence theorem of indecomposable tilting modules when the algebra has a large center and use it to deduce a criterion for an exact functor between two affine highest weight categories to give an equivalence. As an application, we prove that the Arakawa-Suzuki functor [Arakawa-Suzuki, J. of Alg. 209 (1998)] gives a fully faithful embedding of a block of the deformed BGG category of $mathfrak{gl}_{m}$ into the module category of a suitable completion of degenerate affine Hecke algebra of $GL_{n}$.
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 $mathrm{gldim} KQ^{g}/< I^{g}><infty$.
In this paper, we investigate properties of the bounded derived category of finite dimensional modules over a gentle or skew-gentle algebra. We show that the Rouquier dimension of the derived category of such an algebra is at most one. Using this result, we prove that the Rouquier dimension of an arbitrary tame projective curve is equal to one, too. Finally, we elaborate the classification of indecomposable objects of the (possibly unbounded) homotopy category of projective modules of a gentle algebra.
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