Cartier duality

Theorem (Cartier Duality). Let G be a finite abelian group scheme represented by A. Then A° represents another (dual)finite abelian group scheme G°. Here (GDf G, and Hom(G, H) Hom(HD, GD). 

If G and H are any abelian group functors over k, we can always get another group functor Hom(G, H) by attaching to JR the group Hom(GR, Hr). This is the functorial version of Horn, and for H = Gm it is a functorial character group for finite G it is GD. In general it will not be an affine group scheme even when G and H are Cartier duality is one case where it is representable. 

If G is abelian, the product of course is direct. We can also then apply Cartier duality (2.4), because G° need not be connected when G is, and from G° (GD)° x 7t0(GD) we get a corresponding decomposition of GDD G. Applying this to the two factors of G, we get a four-fold decomposition. 

As we have already seen, these results are false in characteristic p explicitly, (o j) xp = x) is an upper triangular copy of Z/pZ. We can however find some restriction on the unipotent groups using Cartier duality. 

Cartan subgroup 77 Cartier duality 17 Center of a group scheme 27 Central simple algebra 145 Character 14 Charater group 55 Clopen set 42 Closed embedding 13 Closed set, closure 156 Closed set in k 28 Closed set in Spec A 42... 

Barsotti-Tate groups. Just as with ordinary Cartier duality, this duality... 

Applying Cartier duality to G, taking the etale quotient of G and applying Cartier duality again we have a filtration ... 

Proof. There is such an F if and only if there is a W lifting Verschiebung on A (because Cartier duality interchanges F and V), hence if and only if A is a canonical lifting. Since by Cartier duality A ( ) =(A(C0)), it... 

In 1959 CARTIER and NISHI proved that for an abelian variety the duality hypothesis holds, i.e. X and are (funotorially) isomorphic (of. Q9], C 10] footnote cf. C31] in C22] this result was used as a hypothesis, see page 216). In this section we show that this result (over arbitrary base preschemes) follows directly from the duality theorem we obtained in section I9. 

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