The derived category of a hereditary order
We give a description of the Auslander-Reiten quiver of the derived category of bounded complexes of lattices for a connected hereditary order \(\Lambda\). Let us assume that \(\Lambda\) has \(n\) non-isomorphic indecomposable lattices. The components of the Auslander-Reiten quiver are of two types: \(\mathbb{Z} A_\infty\) modulo the cyclic group of order \(n\), for each integer \(i\) we have such a component \(T_n (i)\); these correspond to indecomposable complexes of length two with torsion homology. Moreover, there are components of type \(N/\langle n\rangle\); i.e. an oriented sphere with \(n\) vertices. For each integer \(i\) we have such a component \(S_n (i)\). These components correspond to the complexes of lattices, concentrated in one degree. Between the various components there are only morphisms \(T_n (i) \to S_n(i + 1)\) and \(S_n(i) \to T_n(i)\). For each \(i\) the objects in \(S_n(i) \cup T_n (i)\) are in bijection to the indecomposable \(\Lambda\)-modules. Moreover, in the derived category they are isomorphic to the category of \(\Lambda\)-modules. We also show that for the Abelian category of the \(R\)-torsion \(\Lambda\)- modules \(\mathcal T\), the derived category \({\mathcal D}^b({\mathcal T})\) has Auslander-Reiten triangles, and that the Auslander-Reiten quivers of both derived categories coincide. Moreover, we present an example of S. König, which shows that a hereditary order \(\Lambda\) is derived equivalent to a ring \(\Gamma\), which is not an order. This example has the property that 1. Though \(\Lambda\) and \(\Gamma\) are derived equivalent, there does not exist a Rickard-complex, realizing the derived equivalence, 2. If \(F\) is the residue field of \(R\), the \(F\)-algebras \(F\otimes_R\Lambda\) and \(F\otimes_R\Gamma\) are not derived equivalent.
derived category of bounded complexes of lattices
scientific article; zbMATH DE number 815706