A generalization of Rejection Lemma of Drozd-Kirichenko

摘要

Let R be a complete discrete valuation ring, which is fixed once for all as our base ring. Let K denote the quotient field of R. As for basic terminology such as R-lattice, R-order etc., we mostly follow that of [CR]. Let Λ be an R-order in a K-algebra Λ := KΛ approx= K directX_R Λ, and let Ind Λ denote the set of isomorphism classes of indecomposable left Λ-lattices. For an overorder Γ of Λ in Λ, we may naturally consider Ind Γ as a subset of Ind Λ. A subset g of Ind Λ will be called a rejectable subset if there is an overorder Γ such that g = Ind Λ - Ind Γ. The map Γ |→ (Ind Λ — Ind Γ) defines a bijection from the set of all overorders of Λ onto the set of all rejectable subsets of Ind Λ.