Finite flat commutative group schemes over complete discrete valuation rings III: classification, tangent spaces, and semistable reduction of Abelian varieties

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We classify group schemes in terms of their Cartier modules. We also prove the equivalence of different definitions of the tangent space and the dimension for these group schemes; in particular, the minimal dimension of a formal group law that contains S as a closed subgroup is equal to the minimal number of generators for the affine algebra of S. As an application the following reduction criteria for Abelian varieties are proved. Let K be a mixed characteristic local field, let its residue field have characteristic p, L be a finite extension of K, let Oₖₗ be their rings of integers. Let e be the absolute ramification index of L, s=[ₚ(pe/(p-1))], e₀ be the ramification index of L/K, l=2s+vₚ(e₀)+1. For a finite flat commutative Oₗ-group scheme H we denote the Oₗ-dual of the module J/J² by TH. Here J is the augmentation ideal of the affine algebra of H. Let V be an m-dimensional Abelian variety over K. Suppose that V has semistable reduction over L. theor V has semistable reduction over K if and only if for some group scheme H over Oₖ there exist embeddings of Hₖ into Ker[pˡ]ᵥ,ₖ, and of (Oₗ/pˡOₗ)ᵐ into TH. theor This criterion has a very nice-looking version in the ordinary reduction case. theor V has ordinary reduction over K if and only if for some Hₖ⊂ Ker[pˡ]ᵥ,ₖ and M unramified over K we have Hₘ (μₚˡ,ₘ)ᵐ. Here μ denotes the group scheme of roots of unity.theor