On Auslander-Reiten components of string complexes for a certain class of symmetric special biserial algebras

Abstract

Let $\mathbf{k}$ be an algebraically closed field. In this article, inspired by the description of indecomposable objects in the derived category of a gentle algebra obtained by V. Bekkert and H. A. Merklen, we define string complexes for a certain class $\mathcal{C}$ of symmetric special biserial algebras, which are indecomposable perfect complexes in the corresponding derived category. We also prove that if $\Lambda$ is a $\mathbf{k}$-algebra in the class $\mathcal{C}$ and $P^{\bullet }$ is a string complex over $\Lambda$, then $P^{\bullet }$ lies in the rim of its Auslander-Reiten component.