Group schemes of period $$p>2$$.(English)Zbl 1200.14092

Let $$p$$ be a prime number, let $$k$$ be a perfect field of characteristic $$p$$ and let $$W(k)$$ denote the ring of Witt vectors over $$k$$. Let $$K_{00}=\text{Frac}(W(k))$$ and let $$K_0$$ be a totally ramified field extension of $$K_{00}$$ of degree $$e$$. Let $$t$$ be an indeterminate, let $$S=k[[t]]$$ and let $$\sigma: S\rightarrow S$$ denote the map given as $$\sigma(s)=s^p,\forall s\in S$$. Let $$\text{MF}^e_S$$ denote the category of triples $$(M^0,M^1,\varphi_1)$$ where $$M^0$$ is a free $$S$$-module of finite rank, $$M^1$$ is an $$S$$-submodule of $$M^0$$ which satisfies $$t^eM^0\subseteq M^1$$, and $$\varphi_1: M^1\rightarrow M^0$$ is an $$S$$-module morphism which satisfies the conditions $$\varphi_1\sigma=\sigma\varphi_1$$ and $$S\cdot\varphi_1(M^1)=M^0$$.
Next, let $$O_0$$ denote the ring of integers in $$K_0$$. Let $$\text{Gr}_{O_0}'$$ denote the category of finite flat $$p$$-group schemes $$G$$ over the base scheme $$\text{Spec}\;O_0$$. In other words, $$G$$ is a group scheme of the form $$G(-)=\text{Hom}_{O_0\text{-alg}}(H,-)$$ where $$H$$ is a flat commutative unitary $$O_0$$-Hopf algebra of rank a power of $$p$$ over $$O_0$$. Let $$\mu$$, $$\Delta$$ denote multiplication and comultiplication in $$H$$, respectively. For an integer $$l\geq 2$$ define $$\mu^{(l-1)}=\mu(I\otimes \mu)(I\otimes I\otimes \mu)\cdots (I\otimes I\otimes\cdots \otimes I\otimes \mu)$$ ($$l-2$$ copies of $$I$$ in the last factor). Similarily, define $$\Delta^{(l-1)}=(I\otimes I\otimes\cdots \otimes I\otimes \Delta)\cdots (I\otimes I\otimes \Delta)(I\otimes \Delta)\Delta$$. Then there exists an endomorphism of Hopf algebras $$[p]: H\rightarrow H$$ defined as $$[p](h)=\mu^{(p-1)}(\Delta^{(p-1)}(h))$$. By the Yoneda lemma, $$[p]$$ corresponds to an endomorphism of group schemes $$p: G\rightarrow G$$ given as $$p_T(f)=f^p,\forall f\in \text{Hom}_{O_0\text{-alg}}(H,T)$$, where $$T$$ is a commutative unitary $$O_0$$-algebra.
Let $$\epsilon: H\rightarrow O_0$$ denote the counit map of $$H$$ and let $$\lambda: O_0\rightarrow H$$ denote the $$O_0$$-algebra structure map of $$H$$. Then the morphism $$\lambda\epsilon: H\rightarrow H$$ corresponds to a morphism of group schemes $$0: G\rightarrow G$$ (the trivial map). The group $$G$$ is killed by $$p$$ if $$p=0$$. Let $$\text{Gr}_{O_0}$$ denote the full subcategory of $$\text{Gr}_{O_0}'$$ consisting of the group schemes $$G$$ which are killed by $$p$$. The main result of the paper under review shows that there is an antiequivalence between the category $$\text{MF}^e_S$$ and the category $$\text{Gr}_{O_0}$$.

MSC:

 14L15 Group schemes 11G09 Drinfel’d modules; higher-dimensional motives, etc.

Keywords:

Witt vectors; $$p$$-group scheme
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