Finitistic dimension and tilting modules for stratified algebras.
type: Item
instance of: scholarly article
title: Finitistic dimension and tilting modules for stratified algebras.
zbMATH Open document ID: 1073.16008
author: Oleksandr Khomenko
author: Steffen Koenig
author: Volodymyr Mazorchuk
published in: Journal of Algebra
publication date: 2005-05-04T00:00:00Z
review text: The authors prove two general results on the global and finitistic dimension of a quasi-hereditary algebra, and present several corollaries and applications. Let \(A\) be a quasi-hereditary algebra, \({\mathcal F}(\Delta)\) the full subcategory of \(A\)-mod consisting of modules having a costandard filtration, and \(T(A)\) the characteristic tilting module for \(A\). The first result gives sufficient conditions in order that (Theorem 3) \[ \text{gl.dim}(A)=2\dim_{{\mathcal F}(\Delta)}(A)=2\text{p.d.}(T(A)). \] The sufficient conditions are fulfilled when \(A\) is a standard Koszul quasi-hereditary algebra with duality (Corollary 5). The second result, relating global and finitistic dimension is the following: let \(R\) be a complete local commutative algebra over a field \(k\), with maximal ideal \(m\). If \(A\) is a quasi-hereditary algebra over \(R\) and \(I\) is an ideal of \(R\) with finite \(k\)-codimension, then \(A/AI\) is properly stratified, \(A/Am\) is quasi-hereditary and \(\text{fin.dim.}(A/AI)=\text{gl.dim.}(A/Am)\). The two results are applied to calculate the finitistic dimension of several categories of Harish-Chandra bimodules over simple complex finite-dimensional Lie algebras. In particular, the finitistic dimension of a regular block of a thick category \(\mathcal O\) equals twice the projective dimension of the characteristic tilting module in this block. In the final Section, the finitistic dimension of some parabolic generalizations of the BGG category \(\mathcal O\) is computed.
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B55
zbMATH DE Number: 2164791
zbMATH Keywords: finitistic dimension
zbMATH Keywords: global dimension
zbMATH Keywords: stratified algebras
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: quasi-hereditary algebras
zbMATH Keywords: standardly stratified algebras
zbMATH Keywords: characteristic tilting modules
zbMATH Keywords: complex semi-simple finite dimensional Lie algebras
reviewed by: Stefaan Caenepeel
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2005.01.017
OpenAlex ID: W2007618699
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DOI: 10.1016/J.JALGEBRA.2005.01.017
description: scientific article; zbMATH DE number 2164791