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

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.