Algebraic Groups

APPROACH We sketch a more abstract version of the proof of the smoothness of CG.H/. LEMMA 16.24. Let G and H be algebraic groups over k. Let R be a k-algebra, let R0 D R=I with I 2 D 0, and let 0 denote base change R! R0. The obstruction to lifting a homomorphism u0WH0!G0 to R is a class in H .H0;Lie.G0/ ̋I ); if the class is zero, then the set of lifts modulo the action of Ker.G.R/!G.R0// by conjugation is a principal homogeneous space for the group H .H0;Lie.G0/ ̋I /. PROOF. Omitted. 2 LEMMA 16.25. Let H and G be algebraic groups over a ring R, and let R0 D R=I with I 2 D 0. If H is of multiplicative type, then every homomorphism u0WHR0 ! GR0 lifts to a homomorphism uWH ! G; if u is a second lift, then u D inn.g/ ıu for some g 2 Ker.G.R/!G.R0//. PROOF. The cohomology groups H .H0;Lie.G0/ ̋I / and H .H0;Lie.G0/ ̋I / vanish (16.17), and so this follows from (16.24). 2 PROPOSITION 16.26. Let G be an algebraic group over a field k, acting on itself by conjugation, and let H and H 0 be subgroups of G. If G is smooth and H is of multiplicative type, then the transporter TG.H;H / is smooth. PROOF. We use the following criterion (A.53): An algebraic scheme X over a field k is smooth if and only if, for all k-algebras R and ideals I in R such that I 2 D 0, the map X.R/!X.R=I / is surjective. We may replace k with its algebraic closure. Let g0 2 TG.H;H /.R0/. Because G is smooth, g0 lifts to an element g 2G.R/. On the other hand, because H is of multiplicative type, the homomorphism inn.g0/WH0!H 0 0 lifts to a homomorphism uWH !H 0 (see 16.25). The homomorphisms inn.g/WH !G uWH !H 0 ,!G both lift inn.g0/WH0!G0, and so uD inn.g/ı inn.g/ for some g 2G.R/ mapping to e in G.R0/ (see 16.25). Now gg is an element of TG.H;H /.R/ lifting g0. 2 h. Calculation of some extensions 281 COROLLARY 16.27. Let H be a multiplicative algebraic subgroup of an algebraic group G. Then CG.H/ and NG.H/ are smooth. PROOF. This follows from the proposition because NG.H/D TG.H;H/ CG.H/D TG.H;H/: See 1.59 and 1.67. 2 LEMMA 16.28. Let G and H be diagonalizable group varieties and let X be a connected algebraic variety (over an algebraically closed field for simplicity); let WG X !H be a regular map such that x WG!H is a homomorphism for all x 2X.k/; then is constant on X , i.e., factors through the map G X !G.