Relative invariants for representations of finite dimensional algebras

Abstract

 Let M be a finite dimensional module over a finite dimensional basic K-algebra Λ, where K is an algebraically closed field. We associate with M a weight θ^M (i.e. an element of the dual of the Grothendieck group of mod-Λ) in module theoretic terms. Let β be a dimension vector with θ^M(β)=0. We generalize a construction of relative invariants of quivers due to Schofield [S] and define a relative invariant polynomial function d ^M _β on the variety of modules of dimension vector β, such that d ^M _β (N) = 0 for some module N if and only if there is a nonzero morphism from M to N. Assuming char (K) = 0, we conclude from the main result of Schofield-Van den Bergh [SV] that relative invariants of this form span all the spaces of relative invariants. To get algebra generators of the algebra of semi-invariants it is sufficient to take the d ^M _β with M indecomposable.