On Good Filtration Dimensions for Standardly Stratified Algebras

Abstract

Abstract. ∇-good filtration dimensions of modules and of algebras are introduced by Parker for quasi-hereditary algebras. These concepts are now generalized to the setting of standardly stratified algebras. Let A be a standardly stratified algebra. The ∇-good filtration dimension of A is proved to be the projective dimension of the characteristic module of A. Several characterizations of ∇-good filtration dimensions and ∆-good filtration dimensions are given for properly stratified algebras. Finally we give an application of these results to the global dimensions of quasi-hereditary algebras with exact Borel subalgebra. IntroductionAs generalizations of quasi-hereditary algebras, properly stratified algebras and standardly stratifed algebras have been introduced by Cline, Parshall, Scott Recently, in order to calculate the global dimension of the Schur algebra for GL 2 and GL 3 , Parker [13] introduced the notion of ∇-(or ∆-)good filtration dimension for a quasi-hereditary algebra. The aim of this note is to calculate these dimensions for standardly stratified algebras and properly stratified algebras.1991 Mathematics Subject Classification. 16E10, 16G20, 18G20. Key words and phrases. Standardly stratified algebras; properly stratifed algebras; quasi-hereditary algebras; good filtration dimensions; characteristic modules.Research supported by the bilateral project BIL99/43 "New computational, geometric and algebraic methods applied to quantum groups and differential operators" of the Flemish and Chinese governments.

Tilting categories with applications to stratifying systems

“…Then, as a consequence we get interesting results for the theory of stratifying systems. In particular we generalize some of the results obtained by B. Zhu and S. Caenepeel in [5] and also one of the results of V. Mazorchuk and S. Ovsienko in [14]. We also give some examples to illustrate the conditions of the main results.…”

“…Parker introduced in [15] the notion of R Δ-good filtration dimension for quasihereditary algebras. Afterwards, B. Zhu and S. Caenepeel did the same in [5] for standardly stratified algebras. Since, as we have seen above, the notion of stratifying system generalizes the concept of standardly stratified algebra, it would be very interesting to extend such filtration dimension and to get one that makes sense in any algebra.…”

On Characteristic Modules of Graded Quasi-hereditary Algebras

“…Given a quasi-hereditary algebra, of central importance are the standard modules DðlÞ, the D-good module category FðDÞ (Deng and Zhu, 2001;Dlab and Ringel, 1992;Klucznik and Kö nig, 1999;Ringel, 1991;Zhu and Caenepeel, 2004), and in particular, the characteristic module (Ringel, 1991;Zhu and Caenepeel, 2004). For Z-graded or N-graded quasi-hereditary algebras, it was proved in Cline et al (1990) that the standard modules and the defining hereditary chain of a graded quasi-hereditary algebra are gradable.…”

Finitistic dimensions and good filtration dimensions of stratified algegras

“…They appear in the work of Futorny, K6nig and Mazorchuk on a generalisation of the category O ( [7]). Analogously to quasi-hereditary algebras ( [13]), given a standardly stratified algebra (A, A), of central importance are the modules filtered by respectively standard modules, costandard modules, proper standard modules or proper costandard modules (the precise meaning will be given in Section 2) [1,2,12,14,15].…”

“…Recently, in order to calculate the global dimension of the Schur algebra for GLi and GL 3 , Parker [10,11] introduced the notion of V-(or A-)good filtration dimension for a quasi-hereditary algebra. In [15] we generalise this notion to standardly stratified algebras, and observe that the V-good filtration dimension of standardly stratified algebras is the projective dimension of their characterisation module. By using this, we gave an upper bound of the finitistic dimensions for certain class of proper stratified algebras, whose characteristic tilting modules coincide with characteristic cotilting modules.…”