arXiv subjects
Publications and source records attributed to Joseph Winspeare.
In this paper, we give an $A_\infty$-enhancement of the triangulated orbit category $(\mathcal{T}/[1])_{tr}$ when $\mathcal{T}$ is an algebraic triangulated category. We then use this enhancement to give a description of $(\text{per}(A)/[1])_{tr}$ in terms of differential modules and Cohen-Macaulay modules when $A$ is a finite dimensional $\mathbb{Z}$-graded algebra of finite global dimension.
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.