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Rationally Injective Modules for Semisimple Algebraic Groups as Direct Limits

Rationally Injective Modules for Semisimple Algebraic Groups as Direct Limits RATIONALLY INJECTIVE MODULES FOR SEMISIMPLE ALGEBRAIC GROUPS AS DIRECT LIMITS STEPHEN DONKIN In this note we give a short, simple minded proof of the direct limit theorem, a version of which appears as Proposition 2 of [1], as Theorem 6.2 of [3] and appears in §4.7 of [9] . This was conjectured to be true by J. E. Humphreys on the evidence of Winter's treatment of the two dimensional special linear group in [10]. We assume the notation of [8], in particular G denotes a simply connected, semisimple algebraic group over an algebraically closed field K of prime characteristic p, u the m-th hyperalgebra of G for a positive integer m and L(v) the simple rational G module of highest weight v. Let V be a left u module. By a G structure on V we mean a rational G module W such that the restriction of W to u is isomorphic to V. Of course there may not be a G structure on V and even if one does exist it may not be unique (see [4] however). Our theorem asserts that if, for various vel^w e have a G structure Q(y) on Q(l, v) then certain direct http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Rationally Injective Modules for Semisimple Algebraic Groups as Direct Limits

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/12.2.99
Publisher site
See Article on Publisher Site

Abstract

RATIONALLY INJECTIVE MODULES FOR SEMISIMPLE ALGEBRAIC GROUPS AS DIRECT LIMITS STEPHEN DONKIN In this note we give a short, simple minded proof of the direct limit theorem, a version of which appears as Proposition 2 of [1], as Theorem 6.2 of [3] and appears in §4.7 of [9] . This was conjectured to be true by J. E. Humphreys on the evidence of Winter's treatment of the two dimensional special linear group in [10]. We assume the notation of [8], in particular G denotes a simply connected, semisimple algebraic group over an algebraically closed field K of prime characteristic p, u the m-th hyperalgebra of G for a positive integer m and L(v) the simple rational G module of highest weight v. Let V be a left u module. By a G structure on V we mean a rational G module W such that the restriction of W to u is isomorphic to V. Of course there may not be a G structure on V and even if one does exist it may not be unique (see [4] however). Our theorem asserts that if, for various vel^w e have a G structure Q(y) on Q(l, v) then certain direct

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Mar 1, 1980

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