Given a finite dimensional algebra \(A\) over an algebraically closed field and a fixed linear order on the set of isomorphism classes of simple \(A\)-modules there is a classical definition of standard modules associated to this datum. Namely, the standard module corresponding to a fixed indecomposable projective is the quotient of this projective modulo the trace in it of all projectives with strictly bigger indices. The algebra \(A\) is called standardly stratified if projective modules have standard filtration, that is a filtration whose subquotients are standard modules. Standardly stratified algebras are generalizations of quasi-hereditary algebras (in the latter case the additional assumption is that each standard module has trivial endomorphism ring).
In the present paper the author gives several characterizations of algebras which are stratified for all linear orders on the set of isomorphism classes of simple modules. These algebras can be thought of as “stratified analogues” of hereditary algebras. In particular, it turns out that such algebras are directed in the usual sense.
The paper also discusses various properties of the category of modules having standard filtration. In particular, the author classifies all quasi-hereditary algebras for which this category is closed under cokernels of monomorphisms.
Some related results can be found in the unpublished (but available online) preprint of Anders Frisk entitled “On different stratifications of the same algebra” [U.U.D.M. Report 2004:35, Uppsala University (2004)].