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diagonalizable group scheme

A diagonalizable group scheme GG over a ring kk is a group scheme satisfying the following equivalent conditions:

  1. GG is the Cartier dual of a constant group scheme.

  2. O(G)=hom(G,O k)O(G)=hom(G,O_k) is a character group i.e. O(G)O(G) is generated by morphisms Gμ kG\to \mu_k into the multiplicative group scheme.

Proof

Let HH be a constant group scheme. Let G=D(H)G=D(H) be its Cartier dual. By definition we have D(H)(R)=hom Grp(H kR,μ R)hom Grp(H,R ×)hom Alg(k[H],R)D(H)(R)=hom_{Grp}(H\otimes_k R, \mu_R)\simeq hom_{Grp}(H,R^\times)\simeq hom_Alg(k[H],R) and hence G=Speck[H]G=Spec\,k[H]. Here k[H]k[H] is the group algebra of HH and the last isomorphism is given by the adjunction called the universal property of group rings and each ζHk[H]\zeta\in H\subset k[H] is a character of GG. Note that -as is any group algebra- k[H]k[H] is a Hopf algebra.

Conversely, if GG is affine and O(G)O(G) generated by characters, let H be the group of all characters of G; then the canonical map k[H]O(G)k[H]\to O(G) is surjective. But it is always injective (by Dedekind's lemma on linear independence of characters?), hence k[H]O(G)k[H] \simeq O(G).

Revised on June 3, 2012 15:43:35 by Stephan Alexander Spahn (92.106.37.47)