Classification of finite commutative group schemes over complete discrete valuation rings; the tangent space and semistable reduction of Abelian varieties

A complete classification is obtained for finite connected flat commutative group schemes over mixed characteristic complete discrete valuation rings. The group schemes are classified in terms of their Cartier modules. The equivalence of various definitions of the tangent space and the dimension for these group schemes is proved. This shows that the minimal dimension of a formal group law that contains a given connected group scheme $S$ as a closed subgroup is equal to the minimal number of generators for the coordinate ring of $S$. The following reduction criteria for Abelian varieties are deduced.

Suppose $K$ is a mixed characteristic local field with residue field of characteristic $p$, $L$ is a finite extension of $K$, and $\mathfrak{O}_K\subset\mathfrak{O}_L$ are the rings of integers for $K$ and $L$. Let $e$ be the absolute ramification index of $L$, let $s=[\log_p(pe/(p-1))]$, let $e_0$ be the ramification index of $L/K$, and let $l=2s+v_p(e_0)+1$.

For a finite flat commutative $\mathfrak{O}_L$-group scheme $H$, denote by $TH$ the $\mathfrak{O}_L$-dual to $J/J^2$. Here $J$ is the augmentation ideal of the coordinate ring of $H$.

Let $V$ be an $m$-dimensional Abelian variety over $K$. Suppose that $V$ has semistable reduction over $L$.

Theorem (A). $V$ has semistable reduction over $K$ if and only if for some group scheme $H$ over $\mathfrak{O}_K$ there exist embeddings of $H_K$ in $\operatorname{Ker}[p^{l}]_{V,K}$ and of $(\mathfrak{O}_L/p^l\mathfrak{O}_L)^m$ in $TH_{\mathfrak{O}_K}$.

Theorem (B). $V$ has ordinary reduction over $K$ if and only if for some $H_K\subset \operatorname{Ker}[p^{l}]_{V,K}$ and $M$ unramified over $K$ we have $H_M\cong (\mu_{p^{l},M})^m$. Here $\mu$ denotes the group scheme of roots of unity.