Group schemes of period 2. (English) Zbl 1298.14047

The paper under review gives an explicit construction of a functor from the category of finite flat commutative group schemes of period \(2\) defined over a valuation ring of a \(2\)-adic field with algebraically closed residue field to a category of filtered modules satisfying some properties, and shows that the functor is an anti-equivalence.
More specifically, let \(k = \bar{k}\) be a field of characteristic \(2\). Let \(W(k)\) be the ring of Witt vectors and let \(K_{00}\) be the field of fractions of \(W(k)\). Let \(K_0\) be a field extension of \(K_{00}\) of degree \(e\), and let \(O_0\) be the valuation ring of \(K_0\). Let \(\text{Gr}_{O_0}\) be the category of finite flat commutative group schemes \(G\) over \(O_0\) such that \(2 \, \text{id}_G = 0\). For any \(G_1, G_2 \in \text{Gr}_{O_0}\), let \(R(G_1,G_2)\) be the set of morphisms that are of the form \(G_1 \to G_1^{\text{et}} \to G_2^{\text{mult}} \to G_2\) where the first is the natural quotient morphism \(j^{\text{et}} : G_1 \to G_1^{\text{et}}\) and the third morphism is the natural monomorphism \(i^{\text{mult}} : G_2^{\text{mult}} \to G_2\). Let \(\text{Gr}_{O_0}^*\) be the category that has the same objects set as \(\text{Gr}_{O_0}\) and morphism set \(\text{Hom}_{\text{Gr}_{O_0}^*}(G_1,G_2) = \text{Hom}_{\text{Gr}_{O_0}}(G_1,G_2)/R(G_1,G_2)\).
Set \(S = k[[t]]\) where \(t\) is a variable and let \(\sigma : S \to S\) be such that \(\sigma(s) = s^2\). Let \(\text{MF}_S^e\) be the category consisting of the triples \((M^0, M^1, \varphi_1)\) such that \(t^eM^0 \subset M^1 \subset M^0\) are \(S\)-modules, \(M^0\) is a free \(S\)-module of finite rank, \(\varphi_1 : M^1 \to M^0\) is a \(\sigma\)-linear morphism of \(S\)-modules satisfying \(\varphi_1(M^1)S = M^0\). In the category \(\text{MF}_S^e\), each element \({\mathcal M} = (M^0, M^1, \varphi_1)\) has a unique maximal etale subobject \(i^{\text{et}} : {\mathcal M}^{\text{et}} := (M^{0,\text{et}}, t^eM^{1, \text{et}}, \varphi_1) \to {\mathcal M}\), and has a unique maximal multiplicative quotient \(j^{\text{mult}}:{\mathcal M} \to {\mathcal M}^{\text{mult}} := (M^{0,\text{mult}}, M^{1,\text{mult}}, \varphi_1)\). Let \(\text{MF}_S^{e*}\) be the category that has the same objects set as \(\text{MF}_S^e\) and morphism set \(\text{Hom}_{\text{MF}_S^{e*}}({\mathcal M}_1, {\mathcal M}_2) = \text{Hom}_{\text{MF}_S^{e}}({\mathcal M}_1, {\mathcal M}_2)/R({\mathcal M}_1, {\mathcal M}_2)\) where \(R({\mathcal M}_1, {\mathcal M}_2)\) consists of the morphisms of \(\text{MF}_S^e\) of the form \({\mathcal M}_1 \to {\mathcal M}_1^{\text{mult}} \to {\mathcal M}_2^{\text{et}} \to {\mathcal M}_2\). The main theorem of the paper is the following:
Theorem. There is an antiequivalence of categories \[ {\mathcal F}_{O_0}^{O*} : \text{Gr}_{O_0}^* \to \text{MF}_S^{e*}. \]
Reviewer: Xiao Xiao (Utica)


14L15 Group schemes
11S20 Galois theory
14K10 Algebraic moduli of abelian varieties, classification
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