## 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)

### MSC:

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

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