Abstract

This paper continues the program begun by us in [8j2), [9] (see also [15], [18]) in which the authors have begun to exploit in the modular representation theory of semisimple algebraic groups some of the powerful techniques of the theory of derived categories. As noted in the above references, the inspiration for this work comes both from geometry, in the form of the classic algebraic work of Bernstein-Beilinson-Deligne [l] on singular spaces and perverse sheaves, and from the tilting theory of finite dimensional algebras [2], [3], [13], [14]. The present paper broadens and extends this connection with finite dimensional algebra representation theory into a central theme. We begin in Section 1 by completing the results of [9], $ 1, which dealt with “recollement ” of triangulated categories in the sense of [l]. We apply this work in Section 2 to the situation of module categories. While [9], 9 3, treats the case of the natural exact functor Db(mod-B)-+ Db(mod-A) of derived categories arising when B is a quotient ring of A, we consider in this paper the “dual ” situation in which A is a centralizer ring A = End(eB) g eBe, e E B an idempotent. We remark that our interest in this situation was first kindled by Green’s treatment of the Schur algebra in [12], 9 6.

Citations