Approximations of algebras by standardly stratified algebras
Approximations of algebras by standardly stratified algebras
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标准分层代数对代数的逼近
DOI：
10.1016/j.jalgebra.2008.02.017
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
E. Lukács
中科院分区：
文献类型：
--
作者：
I. Ágoston;V. Dlab;E. Lukács
本文试图回答以下问题：给定任意有限维结合K-代数A，是否存在拟遗传代数B，使得所有A-模和所有B-模的子范畴被相应的标准模滤掉后等价.这样的代数称为A的拟遗传逼近。用标准分层代数的适当语言回答了这个问题：对任意K-代数A，存在唯一定义的基本代数B= B（A），使得B B B是Δ-滤子的，且所有Δ-滤子模的子范畴F（ΔA）与F（ΔB）等价;类似地，存在唯一定义的基本代数C=Ω（A），使得CC是Δ <$-filtered的，并且所有Δ <$-filtered的子范畴F（Δ <$A）和F（Δ <$C）都是Δ <$-filtered的。过滤模块是等价的。这些子范畴在分层代数理论中起着基础性的作用。一般来说，由于很难在所有A-模的范畴中局部化这些子范畴，所以构造Ω（A）和Ω（A）常常有助于显式地描述它们。通过对一个代数连续地应用算子Ω和Ω，我们得到了一个标准分层代数序列，经过有限步，它稳定在一个真分层代数中。因此，所有的标准分层代数被划分成（通常是无限的）树，由适当的分层代数索引（作为它们的根）。
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 BBis Δ-filtered and the subcategories F(ΔA) and F(ΔB) of all Δ-filtered modules are equivalent; similarly there is a uniquely defined basic algebra C=Ω(A) such that CCis 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).