By Rotman J.J.

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Let R be a commutative ring, H an R-algebra, C an R-linear Abelian category which is a full subcategory of H - Mod. L; P / ! P is a split monomorphism in R- Mod, for all projective objects P of C. It is not named but the definition is in [40]. 10]. 34. R/ D A- mod. F / D A ˝R F - mod. R/. F /. Finite dimensional Hecke algebras 37 Suppose (i) and (ii). R/. Now, both A and L are free R-modules of finite rank and L is a direct summand of a free A-module of finite rank. L; A/˝R F ! A˝R F is a monomorphism.

Ki C di //, for 1 Ä i Ä d 0 1/. 0 1/. 2 1h/. CŒk; h/. 23]. Ä; h/ is 40 S. CW // factors through the Hecke algebra. CŒk; h/ is a flat CŒk; h-module, by the Künneth spectral sequence. Epq 0 if q ¤ 0. CW / ˝C CŒk; h. 40. Ä; h/’s. CW / ˝C K/ factors through the Hecke algebra. CW / ˝C K. Hence, the result follows. 42. X=W / by the twosided ideal generated by . 0 1/. 0 v1 / . 0 vd 1 / and . i 1/. i C q/, for i ¤ 0. q/ with P ƒ D ƒ0 C diD11 ƒ i . 43. Let R be a Noetherian commutative ring. Let A be a module-finite projective R-algebra, B an R-algebra.

R=. // ! 0 Finite dimensional Hecke algebras 45 and we have a morphism of exact sequences from 0 ! M ! M ! R=. // ! M ! M //. Then, we may show that 0 ! R=. // ! R=. R=. R=. // D 0 and Nakayama’s lemma implies that K D 0. M / and use (a) again. R=. M / ' M . 46(2). 6 Uniqueness of quasihereditary covers. Now we state the uniqueness result in [40]. 48. Suppose that C and C 0 are highest weight categories over R. Let f. / j 2 ƒg and f. 0 / j 0 2 ƒ0 g be standard objects of C and C 0 , respectively.