Tilting categories with applications to stratifying systems

Abstract

In the study of standardly stratified algebras and stratifying systems, we find an object which is either a tilting module or one whose properties strongly remind us of a tilting module. This tilting module appeared already in Dlab and Ringel's work on quasi-hereditary algebras (see [V. Dlab, C.M. Ringel, The module theoretical approach to quasi-hereditary algebras, in: Repr. Theory and Related Topics, in: London Math. Soc. Lecture Note Ser. 168 (1992) 200–224] and [C.M. Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991) 209–223]). Also, this tilting module appears on the work on standardly stratified algebras of I. Ágoston, D. Happel, E. Lukács, and L. Unger, and in the paper of M.I. Platzeck and I. Reiten (see [I. Ágoston, D. Happel, E. Lukács, L. Unger, Standardly stratified algebras and tilting, J. Algebra 226 (2000) 144–160] and [M.I. Platzeck, I. Reiten, Modules of finite projective dimension for standardly stratified algebras, Comm. Algebra 29 (3) (2001) 973–986]). Inspired by them, we introduce the notion of tilting category in order to give a unified approach of these situations for stratifying systems. To do so, we use the ideas of M. Auslander, O. Buchweitz and I. Reiten related to approximation theory (see [M. Auslander, R.O. Buchweitz, The homological theory of maximal Cohen–Macaulay approximations. Mem. Soc. Math. Fr. (N.S.) 38 (1989) 5–37; M. Auslander, I. Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991) 111–152]).