multiplicative group scheme


Let GG be a commutative kk-group functor (in cases of interest this is a finite flat commutative group scheme). Then the Cartier dual D(G)D(G) of GG is defined by

D(G)(R):=Gr R(G kR,μ R)D(G)(R):=Gr_R(G\otimes_k R,\mu_R)

where μ k\mu_k denotes the group scheme assigning to a ring its multiplicative group R ×R^\times consisting of the invertible elements of RR.

This definition deserves the name duality since we have

hom(G,D(H))=hom(H,D(G))=hom(G×H,μ k)hom(G,D(H))=hom(H,D(G))=hom(G\times H,\mu_k)


The notion of a multipliciative group scheme generalizes

Definition and Remark

A group scheme is called multiplicative group scheme if the following equivalent conditions are satisfied:

  1. G kk sG\otimes_k k_s is diagonalizable.

  2. G kKG\otimes_k K is diagonalizable for a field KM kK\in M_k.

  3. GG is the Cartier dual of an étale kk-group.

  4. D^(G)\hat D(G) is an étale kk-formal group.

  5. Gr k(G,α k)=0Gr_k(G,\alpha_k)=0

  6. (If p0)p\neq 0), V GV_G is an epimorphism

  7. (If p0)p\neq 0), V GV_G is an isomorphism


Let G constG_const denote a constant group scheme?, let EE be an étale group scheme?. Then we have the following cartier duals:

  1. D(G const)D(G_const) is diagonalizable.

  2. D(E)D(E) is multiplicative

Last revised on May 4, 2014 at 01:49:41. See the history of this page for a list of all contributions to it.