Finitistic dimension and tilting modules for stratified algebras.
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.
complex semi-simple finite dimensional Lie algebras
scientific article; zbMATH DE number 2164791
scientific article