Classification of discrete derived categories
Let \(A\) be a connected finite-dimensional algebra over an algebraically closed field. The bounded derived category \(\text{D}^b(A{\mathbf mod})\) is said to be discrete [see \textit{D. Vossieck}, J. Algebra 243, 168--176 (2001; Zbl 1038.16010)] if every sequence \((n_i)_{i\in\mathbb Z}\) of non-negative integers occurs as homology dimension vector of at most finitely many indecomposables of \(\text{D}^b(A{\mathbf mod})\). It is known that \(\text{D}^b(A{\mathbf mod})\) is discrete when \(A\) is a path algebra of a Dynkin quiver. In this case, derived equivalence is tantamount to tilting-cotilting equivalence. The non-Dynkin type discrete derived categories \(\text{D}^b(A{\mathbf mod})\) are determined in the present article. More precisely, the authors exhibit a class of algebras \(\Lambda(r,n,m)\) such that \(\text{D}^b(A{\mathbf mod})\) is discrete and not of Dynkin type if and only if \(A\) is derived equivalent (or tilting-cotilting equivalent) to some \(\Lambda(r,n,m)\), and show that the \(r,n,m\) are invariants of the derived equivalence class. Secondly, the Auslander-Reiten components of \(\text{D}^b(A{\mathbf mod})\) (with Auslander-Reiten triangles instead of almost split sequences) are determined for all discrete non-Dynkin \(\text{D}^b(A{\mathbf mod})\). It turns out that only the invariant \(r\) (the number of relations in a certain quiver representation of \(\Lambda(r,n,m)\)) can be read off from the Auslander-Reiten quiver. Therefore, the Euler integral quadratic form and the non-symmetric bilinear homological form of \(\Lambda(r,n,m)\) are also taken into account.
non-Dynkin type discrete derived categories
scientific article; zbMATH DE number 2046451
scientific article