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It turns out that the crucial case is the case r = 1; see [Ja]. 9 Injective GrT-modules We keep the assumptions from Section 8. 1 35 The category of finite dimensional G~T-modules has enough injectives. " Let M be a finite dimensional G~T-module. We have to find a finite dimensional injective GrT-module I containing M. 3). 11(iii). 5)). By Frobenius reciprocity, the identity M ~ M induces a G~T-injection M --~ I. 2 Let A E X(T). Then we set Q~(A)equal to the injective hull of L~(A). 1 (or rather its proof) we have (~(A) C Indra~TL, r(A).

Let M be a finite dimensional G~T-module. We have to find a finite dimensional injective GrT-module I containing M. 3). 11(iii). 5)). By Frobenius reciprocity, the identity M ~ M induces a G~T-injection M --~ I. 2 Let A E X(T). Then we set Q~(A)equal to the injective hull of L~(A). 1 (or rather its proof) we have (~(A) C Indra~TL, r(A). Hence, by injectivity, Q~ (A) is a direct summand of IndTa'TL~ (A). 3 A G

R[G] of finite R[G] is a quotient of a polynomial ring in finitely many variable over R) for which we have G(A) = HomR_alg(R[G], A) for all R-algebras A. We shall call R[G] the coordinate ring of G. Then the definition says that an algebraic group is a representable functor from the category of Ralgebras into the category of groups (with a coordinate ring of finite type). The coordinate ring is a Hopf algebra over R. It clearly determines G uniquely. When R = k is an algebraically closed field, an algebraic group over G(k) = Homk_alg(k[G],k).

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