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Group schemes over Artinian rings and applications. (English) Zbl 1189.14052

Let \(k\) be a perfect field of characteristic \(p\), let \(A=W(k)\) be the ring of Witt vectors over \(k\), and let \(D_k\) be the associated Dieudonné ring. A \(p\)-group scheme over \(k\) is a formal group scheme \(G\) over \(k\) for which \(G\) is isomorphic to the direct limit \(\displaystyle{\lim_{\longrightarrow} G[p^i]}\) where \(G[p^i]\) denotes the kernel of the map \(p^i: G\rightarrow G\), \(i\geq 0\). Let \(h\geq 0\) be an integer. A \(p\)-divisible group scheme over \(k\) of height \(h\) is a \(p\)-group scheme \(G\) for which \(G[p^i]\) has order \(p^{ih}\), \(\forall i\) (equivalently, the representing algebra of \(G[p^i]\) is free of rank \(p^{ih}\) over \(k\)).
There is a categorical equivalence \({\mathcal M}\) between the category of \(p\)-divisible groups over \(k\) and the category of Dieudonné modules, that is, \(D_k\)-modules which are free over \(A\). Given a Dieudonné module \({\mathcal M}(G)\) we can recover the \(p\)-divisible group as follows. From the relations among the Frobenius, \({\mathcal F}\) and the Verschiebung, \({\mathcal V}\) operators, one defines a finite dimensional formal group which in turn produces a \(p\)-divisible group.
Let \(K=\text{Frac}(A)\), let \(L\) be a finite totally ramified extension of \(K\) with ring of integers \(S\). Let \(m\) denote the maximal ideal of \(S\) and put \(S_n=S/m^n\). Choose \(\pi\) so that \(m=(\pi)\) and let \(e\) be so that \((p)=\pi^e\), \(e<p-1\). Observe that \(k\cong A/(m\cap A)\). In one of the main results of this paper (Theorem 2.8), the author shows that there is a categorical duality between the category of \(p\)-divisible groups over \(S_n\) and the category whose objects are certain triples \((L_n,M,\rho)\). In the duality, \(M={\mathcal M}(G_k)\).
Next, suppose that \(k\) is algebraically closed and let \(d,h\) be positive integers which satisfy \(d<h\) and \(\text{gcd}(d,h)=1\). Let \(\Gamma_0\) be a \(p\)-divisible group over \(k\) whose Dieudonné module \({\mathcal M}(\Gamma_0)\) is \(D_k/({\mathcal F}^d-{\mathcal V}^{h-d})\), and is therefore associated to a certain \(p\)-divisible group \(G_{d,h-d}\) over \({\mathbb F}_p\) of dimension \(d\) and height \(h\). One has \(\Gamma_0\cong G_{d,h-d}\times \text{Spec}\;k\). Let \(L\) be a degree \(h\) extension of \({\mathbb Q}_p\) with ring of integers \({\mathcal O}\). Suppose that the ramification index of \(p\) in \(L\) satsifies \(e<p-1\) and suppose that \(S\) is the ring of integers in a totally ramified degree \(e\) extension of \(K\). An \({\mathcal O}\)-lifting of \(\Gamma_0\) is a \(p\)-divisible group \(\Gamma\) over \(S\) so that \(\Gamma_k\cong \Gamma_0\) and \(\text{End}(\Gamma)={\mathcal O}\). In another main result in the paper (Theorem 4.4) the author shows that there exists an \({\mathcal O}\)-lifting of \(\Gamma_0\) if and only if \(h\geq ed\).

MSC:

14L15 Group schemes
14L05 Formal groups, \(p\)-divisible groups
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References:

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