On Standardly Stratified Algebras

Abstract

Let A be a finite dimensional algebra over an algebraically closed field k. For any fixed partial ordering of an index set,Λ say,labelling the simple A-modules L(i),there are standard modules,denoted by Δ(i),i ∈ Λ. By definition,Δ(i) is the largest quotient of the projective cover of L(i) having composition factors L(j) with j ≤ i. Denote by ℱ(Δ) the category of A-modules which have a filtration whose quotients are isomorphic to standard modules. The algebra A is said to be standardly stratified if all projective A-modules belong to ℱ(Δ). In this paper we define a “stratifying system” and we show that this produces a module Y,whose endomorphism ring A is standardly stratified. In particular,we construct stratifying systems for special biserial self-injective algebras.