Anti-equivalence

If So is the spectrum of a perfect field k then classical Dieudonne theory gives us a con-travariant functor G M(G) of C(l)s0 into the category of finite dimensional fc-vectorspaces. Since there is a canonical isomorphism M(GM) = M(G) P the module M(G) is endowed with the action of Frobenius F and Verschiebung V. The functor G (Af(G),F,V) is an anti equivalence of C(l)s0 with the category of triples (Af, F, C ), where M is a finite dimensional fc-vectors pace and F — Af, resp. V M are linear maps such that F o V — 0... 

The above constructions yield an anti-equivalence of categories ... 

Remark 9.4.a) As we are only concerned now with group schemes killed by p we may assume that n = 1. Also the definition of the categories A (l)s, M s and Mns only involve the ring R and do not depend on the ring R or the morphism fs> Thus we could have written M — M So and we could have stated Theorem 9.3 as follows Given R if there exists a ring R as in 9.1 (with n = 1) then there exists an anti-equivalence of categories CrSo... 

It is the reversal of direction which makes us say we have an anti-equivalence. 

Theorem. Separable k-algebras are anti-equivalent to the finite sets on which acts continuously. 

A finite group scheme G over k is called etale if fc[G] is separable. The last theorem shows fc[G] is anti-equivalent to a set X with -action. Also, A k[G] - k[G] k[G] gives a map X x X - X commuting with the <9-action. The dualization here turns the Hopf algebra axioms back into group axioms (see (1.4)). Hence ... 

Theorem. Taking character groups yields an anti-equivalence between group schemes of multiplicative type and abelian groups on which 8 = Gal(k, jk) acts continuously. 

Algebraic affine group scheme 24 Algebraic matrix group 29 Anisotropic torus 56 Anti-equivalence 15 Antipode 8 Arf invariant 147 Artin-Schreier theory 143 Augmentation ideal 13 Automorphism group scheme 58... 

In a natural way D(G) is a 1 A-module, and D is a functor. CARTIER and GABRIEL prove that D is an anti-equivalence between the category of finitely generated A-modules annihilated by some power of V, and the category of unipotent algebraic group schemes over k (k is sup josed to be a perfect field in their theory). Examples ... 

Because the category of commutative formal groups over k is anti equivalent to the category of commutative affine groups over k... 

On dira que G est reflex if (relativement A i) si 1 homemorphisme prgegdent est un isomorphisme. On notera que cela implique que G est commu-tatif. On voit ainsi que le foncteur D induit une anti-equivalence de la... 

Corollaire 7.2. Le foncteur precedent induit une anti-equivalence de la catdgorie des groupes de type multiplicatif et de type fini sur S, et de la categorie des 1 1-modules aui sont de type fini svt Z. ... 

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