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

TL;DR

This paper investigates the injectivity of certain homomorphisms between cohomology groups of formal group schemes deforming the additive and multiplicative groups, revealing conditions under which these maps are injective and describing related dual group schemes.

Contribution

It establishes the injectivity of Frobenius-type homomorphisms on cohomology groups of deformed formal groups over $Z_{(p)}$-algebras and characterizes the duals of their kernels.

Findings

Homomorphism $(psi^{(l)})^*$ is injective under certain restrictions on $lambda$.

The dual of the kernel of $psi^{(l)}$ is a finite group scheme of order $p^l$.

Results apply over $Z_{(p)}$-algebras and describe the structure of associated finite group schemes.

Abstract

Let $\widehat{\mathcal{G}}^{(\lambda)}$ be a formal group scheme which deforms $\widehat{\mathbb{G}}_a$ to $\widehat{\mathbb{G}}_m$. And let $\psi^{(l)}:\widehat{\mathcal{G}}^{(\lambda)}\rightarrow\widehat{\mathcal{G}}^{(\lambda^{p^l})}$ be the $l$-th Frobenius-type homomorphism determined by $\lambda$. We show that the homomorphism $(\psi^{(l)})^\ast:H^2_0(\widehat{\mathcal{G}}^{(\lambda^{p^l})},\widehat{\mathbb{G}}_m)\rightarrow H^2_0(\widehat{\mathcal{G}}^{(\lambda)},\widehat{\mathbb{G}}_m)$ 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.