## J. reine angew. Math. 391 (1988), 85- 99 Journal fur die reine und angewandte Mathematik 0 Walter

### BibTeX

@MISC{Gruyter_j.reine,

author = {De Gruyter and L. Scott At Charlottesville},

title = {J. reine angew. Math. 391 (1988), 85- 99 Journal fur die reine und angewandte Mathematik 0 Walter},

year = {}

}

### OpenURL

### 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.

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Citation Context ...metry, 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 ... |

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Citation Context ..., in terms of the hypercohomology of the algebra A. While in positive characteristic p, the group algebra kS, is of infinite global dimension when plr, the Schur algebra A has finite global dimension =-=[lo]-=- ( see also the discussion in [9], 4. 5d)! In what follows we seek conditions analogous to [9], Theorem 3. 1, which yield the recollement of Section 1 for the bounded derived categories of mod-A and m... |

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Citation Context ...th 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],... |

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Citation Context ...e unifying concept of a highest weight category. Although we obtain this notion by abstracting from the classical representation theory of semisimple groups (or Lie algebras), other examples given in =-=[16]-=-, ?$j 5, 6, indicate that such categories arise in many (perhaps surprising) situations, including quiver algebras and constructible and perverse sheaves. Theorems 3. 4 and 3. 6 relate the theory of h... |

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Citation Context ...Ae has finite (left or right) global dimension by [9], 3.3, 3.4. A similar argument implies fAf has finite global dimension. Conversely, if eAe and fAf have finite global dimensions, A does also (cf. =-=[ll]-=-, Cor. 3. 6, for example). The recollement assertion follows from (1. 3) and (2. 3) since AfA = fA. 0 One can also generalize the “trivial extension” process used in [l l] for constructing triangular ... |

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Citation Context ...roup B (corresponding to the negative roots). Let A =X(T) be the character group of T, partially ordered in the usual way. Fix an integer r >O, and let %? be the category of rational TG,-modules (cf. =-=[S]-=-, [6] for a discussion of the infinitesimal thickenings TG, of T and BG, of B). The irreducible TG,.-modules are indexed by A. For each 1, set A(l) = II;;;. Then %? becomes a highest weight category. ... |

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Citation Context ...al algebras and highest weight categories ‘) By E. Cline at Worcester, B. Pm-shall at Urbana and Charlottesville and L. Scott at Charlottesville 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 categorie... |

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Citation Context ...d highest weight categories ‘) By E. Cline at Worcester, B. Pm-shall at Urbana and Charlottesville and L. Scott at Charlottesville 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 ... |

2 |
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Citation Context ...B (corresponding to the negative roots). Let A =X(T) be the character group of T, partially ordered in the usual way. Fix an integer r >O, and let %? be the category of rational TG,-modules (cf. [S], =-=[6]-=- for a discussion of the infinitesimal thickenings TG, of T and BG, of B). The irreducible TG,.-modules are indexed by A. For each 1, set A(l) = II;;;. Then %? becomes a highest weight category. Simil... |

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Citation Context ... semisimple Lie algebra and let % be the category 0 of Bernstein-Gelfand-Gelfand. Take for A the set of integral weights on a fixed Cartan subalgebra 2 of 3. The category 0 admits a duality functor * =-=[4]-=-, and for each integral weight i, let A(l) = V(A)* be the dual of the Verma module v(1) of high weight 1. In this way, 0 g%” for a highest weight category %7. To verify this, it is sufficient using ar... |

1 |
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Citation Context ... “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. Also, it turns out to lit very well with the stratification theory begun in [9]. In Section 3, we define the unifying concept of a highest weight category. Although we obtain this notion by abs... |

1 | CatCgories d&iv&es, Ctat 0, SGA 4 l/2, Lect. Notes in Math. 569 - Verdier - 1977 |