Approximations of algebras by standardly stratified algebras

Abstract

The paper has its origin in an attempt to answer the following question: Given an arbitrary finite dimensional associative K-algebra A, does there exist a quasi-hereditary algebra B such that the subcategories of all A-modules and all B-modules, filtered by the corresponding standard modules are equivalent. Such an algebra will be called a quasi-hereditary approximation of A. The question is answered in the appropriate language of standardly stratified algebras: For any K-algebra A, there is a uniquely defined basic algebra B = Σ ( A ) such that B B is Δ-filtered and the subcategories F ( Δ A ) and F ( Δ B ) of all Δ-filtered modules are equivalent; similarly there is a uniquely defined basic algebra C = Ω ( A ) such that C C is Δ ¯ -filtered and the subcategories F ( Δ ¯ A ) and F ( Δ ¯ C ) of all Δ ¯ -filtered modules are equivalent. These subcategories play a fundamental role in the theory of stratified algebras. Since, in general, it is difficult to localize these subcategories in the category of all A-modules, the construction of Σ ( A ) and Ω ( A ) often helps to describe them explicitly. By applying consecutively the operators Σ and Ω for an algebra, we get a sequence of standardly stratified algebras which, after a finite number of steps, stabilizes in a properly stratified algebra. Thus, all standardly stratified algebras are partitioned into (generally infinite) trees, indexed by properly stratified algebras (as their roots).