Singularity categories of gentle algebras

Abstract

We determine the singularity category of an arbitrary finite‐dimensional gentle algebra normalΛ. It is a finite product of n‐cluster categories of type A1. Equivalently, it may be described as the stable module category of a self‐injective gentle algebra. If normalΛ is a Jacobian algebra arising from a triangulation MJX-tex-caligraphicscriptT of an unpunctured marked Riemann surface, then the number of factors equals the number of inner triangles of MJX-tex-caligraphicscriptT.

On the Recollements of Functor Categories

“…This algebra is gentle, see [21] for definition. So, by [31,Theorem 2.4] the indecomposable Gorenstein projective A-modules are represented as follows.…”

On Extensions for Gentle Algebras

“…We note that gentle algebras are string algebras and that there is a large body of work on string algebras. In particular, projective resolutions and syzygies, have been considered before, see for example [15,16]. In [24], minimal projective presentations of string and band modules were given in terms of string combinatorics, which in the case of gentle algebras can be formulated in terms of homotopy string combinatorics.…”

“…The projective resolutions of indecomposable modules over gentle algebras are well understood, see, for example, [16]. So it is surprising that up to now, in general, no complete combinatorial description of the extensions between indecomposable modules over a gentle algebra is known.…”

On syzygies over 2-Calabi–Yau tilted algebras

“…In [23], the author computes CMP(A) where A = kQ/I is a gentle algebra. The indecomposable modules in CMP(A) are given by the non-projective indecomposable summands of radP (a), where a is a vertex in a cycle α 1 .…”

“…In this subsection we describe CMP(B) for the disk with one puncture, or equivalently, for cluster-tilted algebras of Dynkin type D. This gives an answer to [23,Remark 3.7] in this case. This situation is much more complicated than the unpunctured case, and we will need to distinguish between 3 types of triangulations.…”