A Geometric Model For The Module Category Of Skew-gentle Algebras

Skew-gentle algebras,as a special kind of clannish algebras,has important applica-tions in the study of cluster categories for marked surface with punctures.In that case,a triangulation on the punctured marked surface can give rise to a skew-gentle algebra.And the properties of such algebra will translate the result of cluster category to the topo-logical result on the surface.On the other hand,for an arbitrary skew-gentle algebra,how to embed it into a punctured marked surface,transfer it into the partial triangulation on the surface,and give a geometric description of its module category is important and un-known.In this article,skew-gentle algebras are realised as skew-tiling algebras,which are associated to partial triangulations of marked surfaces with punctures.The important result on the module category of skew-gentle algebra will be given by the curves on the punctured marked surface.We will use this description to give a geometric model of the module category of any skew-gentle algebras.