Properly stratified endomorphism algebras.
The authors prove the following statement: Theorem. Let \(R\) be a commutative, local, self-injective algebra over an algebraically closed field. Let \({\mathcal X}=\{X(1)=R,X(2),\dots,X(n)\}\) be a sequence of local-colocal \(R\)-modules reflecting inclusions of the corresponding local ideals \(\underline X(i)\) of \(R\) (i.e. \(\underline X(i)\subset\underline X(j)\) implies \(j\leq i\)). Let \(X=\bigoplus_{i=1}^nX(i)\) and \(A=\text{End}_R(X)\). Then the algebra \(A\) is properly stratified with respect to the natural order on \(\{1,\dots,n\}\) if and only if: (1) \(\underline X(i)\cap\underline X(j)\) is generated by suitable \(\underline X(t)\) for all \(i,j\); and (2) \(\underline X(j)\cap(\sum_{t=j+1}^n\underline X(t))=\sum_{t=j+1}^n(\underline X(j)\cap\underline X(t))\) for all \(j\).
scientific article; zbMATH DE number 2153353