Characteristic modules of quasi-hereditary algebras with three simples
The main result of this note is the following: Let \(T\) be a generalized tilting module over a basic quasi-hereditary algebra \((A,\leq)\) with three simple modules and suppose that the indecomposable direct summands of \(T\) satisfy two conditions (to have unique `highest weight' and to be modules over certain quasi-hereditary quotient algebras of \(A\)) which are well-known to hold true for the characteristic tilting module of \(A\). Then \(T\) actually must be the characteristic tilting module of \((A,\leq)\). Reviewer's remark: The same result holds true without any assumption on the number of \(A\)-simples. This can be shown (by induction on the number of simples) using lemma 4 in the paper and the fact, that computing cohomology one may pass from \(A\) to a certain quasi-hereditary quotient algebra if the modules involved are defined over this quotient algebra.
scientific article; zbMATH DE number 1077938