Approximations of algebras by standardly stratified algebras.
Let \(A\) be a finite-dimensional associative algebra with a fixed (natural) linear order on the set \(e_1,\dots,e_n\) of primitive orthogonal idempotents. For every \(i=1,\dots,n\) we define the standard module \(\Delta(i)\) as the quotient of \(Ae_i\) modulo the trace of all \(Ae_j\), \(j>i\). The proper standard module \(\overline\Delta(i)\) is defined as the quotient of \(\Delta(i)\) modulo the trace of \(Ae_i\) in the radical of \(\Delta(i)\). Denote by \(\mathcal F(\Delta,A)\) and \(\mathcal F(\overline\Delta,A)\) the full subcategories of \(A\)-mod, consisting of all modules which have a filtration, whose subquotients are standard or proper standard modules, respectively. The principal question addressed in the paper under review is: Given some algebra \(A\), does there exist a quasi-hereditary algebra \(B\) such that the subcategories \(\mathcal F(\Delta,A)\) and \(\mathcal F(\Delta,B)\) are equivalent? The authors show that for any \(A\) there is a unique basic algebra \(\Sigma(A)\) such that \(\Sigma(A)\) is standardly stratified and the categories \(\mathcal F(\Delta,A)\) and \(\mathcal F(\Delta,\Sigma(A))\) are equivalent. Further, there is a unique basic algebra \(\Omega(A)\) such that \(\Omega(A)^{op}\) is standardly stratified and the categories \(\mathcal F(\overline\Delta,A)\) and \(\mathcal F(\overline\Delta,\Sigma(A))\) are equivalent. It is further shown that the algebra \((\Omega\Sigma)^{n-1}(A)\) is always properly stratified. The paper contains several interesting examples.
scientific article; zbMATH DE number 5304643