On the injectivity of certain homomorphisms between extensions of ${\hat{\mathcal{G}}}^{(\lambda )}$ by ${\hat{\mathbb{G}}}_m$ over a ${\mathbb{Z}}_{(p)}$-algebra

Abstract

Let ${\hat{\mathcal{G}}}^{(\lambda )}$ be a formal group scheme which deforms ${\hat{\mathbb{G}}}_a$ to ${\hat{\mathbb{G}}}_m$. And let ${\psi }^{(l)}:{\hat{\mathcal{G}}}^{(\lambda )}\to {\hat{\mathcal{G}}}^{({\lambda }^{p^l})}$ be the $l$-th Frobenius-type homomorphism determined by $\lambda$. We show that the homomorphism $({\psi }^{(l)}{)}^{\ast }:H_0^2({\hat{\mathcal{G}}}^{({\lambda }^{p^l})},{\hat{\mathbb{G}}}_m)\to H_0^2({\hat{\mathcal{G}}}^{(\lambda )},{\hat{\mathbb{G}}}_m)$ induced by ${\psi }^{(l)}$ is injective over a ${\mathbb{Z}}_{(p)}$-algebra under a suitable restriction on $\lambda$. In this situation, the Cartier dual of $\mathrm{Ker}({\psi }^{(l)})$, which is a finite group scheme of order $p^l$, is described over a $\mathbb{Z}/(p^n)$-algebra.

16 pages