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Approximations of algebras by standardly stratified algebras
- Source :
-
Journal of Algebra . May2008, Vol. 319 Issue 10, p4177-4198. 22p. - Publication Year :
- 2008
-
Abstract
- 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 such that is Δ-filtered and the subcategories and of all Δ-filtered modules are equivalent; similarly there is a uniquely defined basic algebra such that is -filtered and the subcategories and of all -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 and 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). [Copyright &y& Elsevier]
- Subjects :
- *MATHEMATICS
*ALGEBRA
*MATHEMATICAL analysis
*MODULES (Algebra)
Subjects
Details
- Language :
- English
- ISSN :
- 00218693
- Volume :
- 319
- Issue :
- 10
- Database :
- Academic Search Index
- Journal :
- Journal of Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 31675320
- Full Text :
- https://doi.org/10.1016/j.jalgebra.2008.02.017