The 1-periodic derived category of a gentle algebra : Part 1 -- Indecomposable objects

Abstract

Combining results from Keller and Buchweitz, we describe the 1-periodic derived category of a finite dimensional algebra $A$ of finite global dimension as the stable category of maximal Cohen-Macaulay modules over some Gorenstein algebra $A^\ltimes$. In the case of gentle algebras, using the geometric model introduced by Opper, Plamondon and Schroll, we describe indecomposable objects in this category using homotopy classes of curves on a surface. In particular, we associate a family of indecompoable objects to each primitive closed curve. We then prove using results by Bondarenko and Drozd concerning a certain matrix problem, that this constitutes a complete description of indecomposable objects.