Derived tame tree algebras

Christof Geiss

Abstract: A tree algebra is a finite dimesnional algebra, where the underlying graph of the Gabriel quiver is a tree. Our main result states, that a tree algebra is derived tame iff its Euler form is nonegative. (The same result was obtained independly by Th. Bruestle with different methods). Our strategy is based on induction on the number of vertices of the quiver: Suppose the result is true for A, and M is an indecomposable A-module, then we have to show, that the one-point extension A[M] is derived tame iff the Euler form of A[M] is nonegative. This is well-known in case A is piecewise hereditatary, otherwise A is derived equivalent to a special kind of clannish algebra, a so called semi-chain. In order to continue the induction we need the following: - Discuss the notion of "derived tameness" - Understand the "derived subspace problem", i.e. how is D^b(A[M]) obtained from D^b(A) and RHom(M,-) - Describe the derived category of the semichain-algebras (dimension vectors of indecomposables, hommorphisms between indecomposables).

created Fri Mar 17 16:36:04 PST 2000