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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.
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