Standardly Stratified Algebras and Tilting

Abstract

The concept of the characteristic tilting module and of the Ringel dual for quasihereditary algebras is generalized for the setting of standardly stratified algebras.

On Good Filtration Dimensions for Standardly Stratified Algebras

“…The covariant finiteness of T , left T -approximations of M and the minimal left T -approximation of M can be defined using duality arguments (compare [1], [15]). …”

“…They appear in the work of Futorny, König and Mazorchuk on a generalization of the category O [8]. Analogously to quasi-hereditary algebras [2], [4], [17], given a standardly stratified algebra (A, Λ), of central importance are the modules filted by respectively standard modules, costandard modules, proper standard modules or proper costandard modules (the precise meaning will be given in Section 1) [1], [15]. 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.…”

Relative theory in subcategories

“…By induction and using (1) and the resolution of P , we find that the sequence 0 Ñ pZ n¡k , M i 1 q Ñ pZ n¡k , T i q Ñ pZ n¡k , M i q Ñ 0 is exact for i ≥ r 1 k and 0 k ≤ n. In particular, for k n, the sequence 0 Ñ M i 1 Ñ T i Ñ M i Ñ 0 is exact for i ≥ r n 1. Then by (1) it is F -exact.…”

A construction of strong tilting modules

“…It has been shown in [16] (more generally in [1,15]) that the subcategory F( ) ∩ F(∇) = add T , where T is a tilting -module known as the characteristic tilting module. Moreover, T is uniquely defined and has the following properties: [1,15] Let ( , ≤) be a standardly stratified algebra and T the characteristic tilting module. Then…”

“…We first recall some definitions, notations and results from [1,3,5,6,15,16]. For unexplained terminologies we refer the reader to [2,4,17].…”