 
Summary: APPROXIMATIONS OF ALGEBRAS
BY STANDARDLY STRATIFIED ALGEBRAS
Istv´an ´Agoston1
, Vlastimil Dlab2
and Erzs´ebet Luk´acs1
Abstract. The paper has its origin in an attempt to answer the following question:
Given an arbitrary finite dimensional associative Kalgebra A, does there exist a quasi
hereditary algebra B such that the subcategories of all Amodules and all Bmodules,
filtered by the corresponding standard modules are equivalent. Such an algebra will
be called a quasihereditary approximation of A. The question is answered in the
appropriate language of standardly stratified algebras: For any Kalgebra A, there
is a uniquely defined basic algebra B = (A) such that BB is filtered and the
subcategories F(A) and F(B) of all filtered modules are equivalent; similarly
there is a uniquely defined basic algebra C = (A) such that CC is filtered and
the subcategories F(A) and F(C ) 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 Amodules,
the construction of (A) and (A) often helps to describe them explicitly. By applying
consecutively the operators and for an algebra, we get a sequence of standardly
stratified algebras which, after a finite number of steps, stabilizes in a properly stratified
