Successive extensions of minimal \(p\)-divisible groups
Let \(K\) be a field, let \(q\in {\mathbb Z}\) be prime, let \(T\) be an indeterminate, and let \(K[\overline T]=K[T]/(T^q-1)\). Then \(K[\overline T]\) is a \(K\)-Hopf algebra of dimension \(q\) with comultiplication \(\Delta: K[\overline T]\rightarrow K[\overline T]\otimes K[\overline T]\) defined by \(\overline T\mapsto\overline T\otimes \overline T\). Thus \(\mu_q=\text{Spec}\;K[\overline T]=\text{Hom}_{K-alg}(K[\overline T],-)\) is a representable group functor from the category of commutative \(K\)-algebras to the category of abelian groups. For a commutative \(K\)-algebra \(A\), \(\mu_q(A)\) consists of functions \(f_a: K[\overline T]\rightarrow A\) with \(\overline T\mapsto a\), \(a^q=1\). The group product in \(\mu_q(A)\) is defined as \((f_a*f_b)(T)=(f_a,f_b)\Delta(\overline T)=f_a(T)f_b(T)=ab\). We assume that \(K\) is a perfect field of characteristic \(p\) (with \(p\) not necessarily \(= q\)). Then the map \(\phi: K\rightarrow K\), \(r\mapsto r^p\), is an automorphism. Viewing \(K\) as a \(K\)-module via \(r\cdot s=\phi(r)s=r^ps\), one defines the group scheme \(\mu_q^{(p)}= \text{Spec}\;(K[\overline T]\otimes_K K)\), called the twist of \(\mu_q\). Since \(K\) has characteristic \(p\), \({\mathcal F}: K[\overline T]\otimes_K K\rightarrow K[\overline T]\) defined as \({\overline T\otimes 1}\rightarrow {\overline T}^p\) is an algebra map (in fact, it is a Hopf algebra map), and so by Yoneda's Lemma, \({\mathcal F}\) corresponds to a endomorphism of group schemes \(F_{\mu_q}: \mu_q\rightarrow \mu_q^{(p)}\). This is the Frobenius homomorphism on \(\mu_q\). One also has a shift homomorphism \(V_{\mu_q}: \mu_q^{(p)}\rightarrow \mu_q\) which satisfies \(V_{\mu_q}\circ F_{\mu_q}=[p]_{\mu_q}\) and \(F_{\mu_q}\circ V_{\mu_q}=[p]_{\mu_q^{(p)}}\) where \([p]_{\mu_q}\), \([p]_{\mu_q^{(p)}}\) are the \(p\)th power maps on \(\mu_q\), \(\mu_q^{(p)}\). In \(\mu_q(A)\), one has \([p](f_a)({\overline T})=F_{\mu_q}(f_a)({\overline T})\). Thus \(F_{\mu_q}=[p]_{\mu_q}\) and \(V_{\mu_q}=\text{Id}_{\mu_q}\). Generally, let \(G\) be a finite commutative group scheme over \(K\) and let \(F_G: G\rightarrow G^{(p)}\) and \(V_G: G^{(p)}\rightarrow G\) denote the Frobenius and shift homomorphisms. If \(\text{Im}(V_G: G^{(p)}\rightarrow G)=\text{Ker}(F_G: G\rightarrow G^{(p)})\) and \(\text{Im}(F_G: G\rightarrow G^{(p)})=\text{Ker}(V_G: G^{(p)}\rightarrow G)\), then \(G\) is a Barsotti-Tate group of level one (\(BT_1\)). For example, if \(p=q\), then \(\mu_q\) is \(BT_1\). Let \(W(K)\) denote the ring of Witt vectors, and let \(\sigma: W(K)\rightarrow W(K)\) denote the Frobenius map. The Dieudonné ring is the non-commutating polynomial ring \(A_K=W(K)[{\mathcal F},{\mathcal V}]\) modulo the conditions \({\mathcal F}x=x^\sigma{\mathcal F}\), \({\mathcal V}x^\sigma=x{\mathcal V}\), \({\mathcal F}{\mathcal V}={\mathcal V}{\mathcal F}=p\), \(\forall x\in W(K)\). Observe that in the case \(K={\mathbb F}_p\), \(G=\mu_q\), \(q\not = p\), with \({\mathcal F}\), \({\mathcal V}\) thought of as the Hopf algebra maps on the appropriate coordinate rings, the Dieudonné ring is \({\mathbb Z}_p[{\mathcal F}]/({\mathcal F}-p)={\mathbb Z}_p\). If \(q=p\), then the Dieudonné ring is \({\mathbb F}_p\). A Dieudonné module is a left \(A_K\)-module which is finitely generated over \(W(K)\). Let \({\mathbb G}=(G_n,i_n)_{n\geq 1}\) be a \(p\)-divisible group. If we consider \({\mathbb G}\) as a \(p^\infty\)-torsion group scheme, the \(p^n\)-kernel of \({\mathbb G}\) is \({\mathbb G}[p^n]=G_n\). For example, \({\mathbb M}=(\mu_{p^n},i_n)_{n\geq 1}\) is a \(p\)-divisible group and \({\mathbb M}[p]=\mu_p\). There is an equivalence \({\mathcal D}\) between the category of \(p\)-divisible groups and the category of Dieudonné modules. For example, in case \(K={\mathbb F}_p\), \({\mathcal D}({\mathbb M})={\mathbb F}_p\). The main theorem of the paper under review (Theorem 1.1) states that given a \(BT_1\) \(G\) over an algebraically closed field of characteristic \(p\), there exists an injective homomorphism \(\Psi: H_{em,en}[p]\rightarrow G\) where \(H_{em,en}\) is a \(p\)-divisible group determined by certain integers \(e,m,n\). \(H_{em,en}\) is constructed by choosing a suitable Dieudonné module and then employing \({\mathcal D}^{-1}\). Theorem 1.1 is then used to characterize \(G\) in terms of minimal \(BT_1\)s. Note that with \(e=1,m=0,n=1\), we have \(H(0,1)\cong {\mathbb M}\), and in this case, \(\Psi: H(0,1)[p]\rightarrow\mu_p\) is an isomorphism, and so, \(\mu_p\) is a minimal \(BT_1\).
scientific article; zbMATH DE number 5561734