A geometric model for the module category of a skew-gentle algebra
Ping He, Yu Zhou, Bin Zhu
备注：
Presentation was improved, two tables of notations were added, and a correction for the last condition in the definition of tagged permissible curves was made
In this article, we realize skew-gentle algebras as skew-tiling algebras associated to admissible partial triangulations of punctured marked surfaces. Based on this, we establish a bijection between tagged permissible curves and certain indecomposable modules, interpret the Auslander-Reiten translation via the tagged rotation, and show intersection-dimension formulas. As an application, we classify support $τ$-tilting modules via maximal collections of non-crossing tagged generalized permissible curves.
Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of orbifold surfaces establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their sin…
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 $\mathrm{gldim} KQ^{g}/< I^{g}><\infty$.
Skew-gentle algebras are skew-group algebras of gentle algebras equipped with a certain $\Z_2$-action. Building on the bijective correspondence between gentle algebras and dissected surfaces, we obtain in this paper a bijection between skew-gentle algebras and certain dissected orbifolds that admit a double cover. We prove the compatibility of the $\Z_2$-action on the double cover with the skew-group algebra construction. This allows us to investigate the derived equivalence relation between skew-gentle algebras in geometric terms: We associate to each skew-gentle algebra a line field on the o…
This is an extended abstract of my talk at the Oberwolfach-Workshop "Representation Theory of Finite-Dimensional Algebras" (19-24 January 2020). It introduces a geometric orbifold model for the bounded derived category of a skew-gentle algebra. This is joint work with Daniel Labardini-Fragoso and Yadira Valdivieso-Díaz.
Skew-gentle algebras are skew-group algebras of certain gentle algebras endowed with a Z 2-action. Using the topological description of Opper, Plamondon and Schroll in [OPS] for the indecomposable objects of the derived category of any gentle algebra, one obtains here a complete description of indecomposable objects in the derived category of any skew-gentle algebras in terms of curves on an orbifold surface.The results presented here are complementary to the ones in [LSV]. First, we obtain a complete classification of indecomposable objects and not of ''homotopy strings'' and ''homotopy bands…
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.
In this note we correct two oversights in [Mapping cones in the bounded derived category of a gentle algebra, J. Algebra 530 (2019), 163--194, also arXiv:1609.09688] which only occur when a band complex is involved. As a consequence we see that the mapping cone of a morphism between two band complexes can decompose into arbitrarily many indecomposable direct summands.
Bongartz and Ringel proved that there is no gaps in the sequence of lengths of indecomposable modules for the finite-dimensional algebras over algebraically closed fields. The present paper mainly study this "no gaps" theorem for the bounded derived module category $D^b(A)$ of a gentle algebra $A$: if there is an indecomposable object in $D^b(A)$ of cohomological length $l>1$, then there exists an indecomposable with cohomological length $l-1$.
Let $\mathcal{D}$ be a Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category with a rigid object $R$. Let $Λ=\operatorname{End}_{\mathcal{D}}R$ be the endomorphism algebra of $R$. We introduce the notion of mutation of maximal rigid objects in the two-term subcategory $R\ast R[1]$ via exchange triangles, which is shown to be compatible with mutation of support $τ$-tilting $Λ$-modules. In the case that $\mathcal{D}$ is the cluster category arising from a punctured marked surface, it is shown that the graph of mutations of support $τ$-tilting $Λ$-modules is isomorphic to the graph of fli…
We determine the singularity category of an arbitrary finite dimensional gentle algebra $Λ$. It is a finite product of $n$-cluster categories of type $\mathbb{A}_{1}$. Equivalently, it may be described as the stable module category of a selfinjective gentle algebra. If $Λ$ is a Jacobian algebra arising from a triangulation $\ct$ of an unpunctured marked Riemann surface, then the number of factors equals the number of inner triangles of $\ct$.
In this article we provide a simple combinatorial description of morphisms between indecomposable complexes in the bounded derived category of a gentle algebra.
Cartan matrices are of fundamental importance in representation theory. For algebras defined by quivers (i.e. directed graphs) with relations the computation of the entries of the Cartan matrix amounts to counting nonzero paths in the quivers, leading naturally to a combinatorial setting. In this paper we study a refined version, so-called q-Cartan matrices, where each nonzero path is weighted by a power of an indeterminant q according to its length. Specializing q=1 gives the classical Cartan matrix. Our main motivation are derived module categories and their invariants: the invariant factors…
A gentle algebra gives rise to a dissection of an oriented marked surface with boundary into polygons and the bounded derived category of the gentle algebra has a geometric interpretation in terms of this surface. In this paper we study silting theory in the bounded derived category of a gentle algebra in terms of its underlying surface. In particular, we show how silting mutation corresponds to the changing of graded arcs and that in some cases silting mutation results in the interpretation of the octahedral axioms in terms of the flipping of diagonals in a quadrilateral as in the work of Dyc…
In this article, gentle algebras are realised as tiling algebras, which are associated to partial triangulations of unpunctured surfaces with marked points on the boundary. This notion of tiling algebras generalise the notion of Jacobian algebras of triangulations of surfaces and the notion of surface algebras. We use this description to give a geometric model of the module category of any gentle algebra.
Let $A$ be a graded algebra. It is shown that the derived category of dg modules over $A$ (viewed as a dg algebra with trivial differential) is a triangulated hull of a certain orbit category of the derived category of graded $A$-modules. This is applied to study derived categories of graded gentle one-cycle algebras.
In this article we describe the triangulated structure of the bounded derived category of a gentle algebra by describing the triangles induced by the morphisms between indecomposable objects in a basis of their Hom-space.
In this article we describe indecomposable objects of the derived categories of a branch class of associative algebras. To this class belong such known classes of algebras as gentle algebras, skew-gentle algebras and certain degenerations of tubular algebras.
In this paper, using the correspondence of gentle algebras and dissections of marked surfaces, we study full exceptional sequences in the perfect derived category $\mathsf{K^b(A)}$ of a gentle algebra $\mathsf{A}$. We show that full exceptional sequences in $\mathsf{K^b(A)}$ exist if and only if the associated marked surface has no punctures and has at least two marked points on the boundary. Furthermore, by using induction on cuts of surfaces, we characterise when an exceptional sequence can be completed to a full exceptional sequence. If the genus of the associated surface is zero then we sh…
We give a complete description of a basis of the extension spaces between indecomposable string and quasi-simple band modules in the module category of a gentle algebra.
We investigate the role of gentle algebras in higher homological algebra. In the first part of the paper, we show that if the module category of a gentle algebra $Λ$ contains a $d$-cluster tilting subcategory for some $d \geq 2$, then $Λ$ is a radical square zero Nakayama algebra. This gives a complete classification of weakly $d$-representation finite gentle algebras. In the second part, we use a geometric model of the derived category to prove a similar result in the triangulated setup. More precisely, we show that if $\mathcal{D}^b(Λ)$ contains a $d$-cluster tilting subcategory that is clos…
We consider the homotopy category of complexes of projective modules over any gentle algebra. We prove that indecomposable $Σ$-pure-injective objects in s must be shifts of string or band complexes. We begin with a survey of purity in compactly generated triangulated categories, recalling some characterisations of $Σ$-pure-injective objects that mimic classical results from the model theory of modules. We then specify to the aforementioned homotopy category, and describe the compact objects in this case. Our proof uses a recent adaptation of a classification technique known as the functorial f…
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 $τ$-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, w…
We determine the derived representation type of Nakayama algebras and prove that a derived tame Nakayama algebra without simple projective module is gentle or derived equivalent to some skewed-gentle algebra, and as a consequence, we determine its singularity category.
We consider algebras defined over a complete, local and noetherian ground ring. They are gentle algebras in case the ground ring is a field. The unbounded homotopy category of complexes of projective modules is considered. Complexes with finitely-generated homogeneous components are shown to be isomorphic to direct sums of indecomposable string and band complexes. The corresponding isoclasses are described, and the Krull-Remak-Schmidt-Azumaya property is verified. This classification problem is solved using the idea of functorial filtrations.
In this paper we refine the main result of a previous paper of the author with Grimeland on derived invariants of surface algebras. We restrict to the case where the surface is a torus with one boundary component and give an easily computable derived invariant for such surface algebras. This result permits to give answers to open questions on gentle algebras: it provides examples of gentle algebras with the same AG-invariant (in the sense of Avella-Alaminos and Geiss) that are not derived equivalent and gives a partial positive answer to a conjecture due to Bobiński and Malicki on gentle $2$-c…
Let $\mathbf{S}$ be a graded marked surface. We construct a string model for Calabi-Yau-$\mathbb{X}$ category $\mathcal{D}_\mathbb{X}(\mathbf{S}_\bigtriangleup)$, via the graded DMS (=decorated marked surface) $\mathbf{S}_Δ$. We prove an isomorphism between the braid twist group of $\mathbf{S}_\bigtriangleup$ and the spherical twist group of $\mathcal{D}_\mathbb{X}(\mathbf{S}_\bigtriangleup)$, and $\mathbf{q}$-intersection formulas. We also give a topological realization of the Lagrangian immersion $\mathcal{D}_\infty(\mathbf{S})\to\mathcal{D}_\mathbb{X}(\mathbf{S}_\bigtriangleup)$, where $\ma…
Following the approach of Haiden-Katzarkov-Kontsevich arXiv:1409.8611, to any homologically smooth graded gentle algebra $A$ we associate a triple $(Σ_A, Λ_A; η_A)$, where $Σ_A$ is an oriented smooth surface with non-empty boundary, $Λ_A$ is a set of stops on $\partial Σ_A$ and $η_A$ is a line field on $Σ_A$, such that the derived category of perfect dg-modules of $A$ is equivalent to the partially wrapped Fukaya category of $(Σ_A, Λ_A ;η_A)$. Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of $Σ_A$ on the h…
In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beattie's construction of a ring with local units in \cite{be}. We recover in a categorical generalized setting the Duality Theorems of M. Cohen and S. Montgomery in \cite{cm}, and we provide a unification wit…
