ON THE INJECTIVITY OF CERTAIN HOMOMORPHISMS BETWEEN HOCHSCHILD COHOMOLOGIES RELATED TO ALGEBRAIC TORI

Abstract

Let $\hat{\mathcal{G}}^{(\lambda)}$ be a formal group scheme which deforms $\hat{\mathbb{G}}_a$ to $\hat{\mathbb{G}}_m$, and let $\hat{\mathcal{E}}^{(\lambda, \mu ; D)}$ be an extension of $\hat{\mathcal{G}}^{(\lambda)}$ by $\hat{\mathcal{G}}^{(\mu)}$. Suppose
$$
\psi^{(l)}: \hat{\mathcal{E}}^{(\lambda, \mu ; D)} \rightarrow \hat{\mathcal{E}}^{\left(\lambda^{p^l} ; \mu^{p^l} ; D^{\prime}\right)}
$$
is an $l$ th Frobenius-type homomorphism, which is determined by $\lambda$ and $\mu$. Then, we show that the homomorphism
$$
\left(\psi^{(l)}\right)^*: H_0^2\left(\hat{\mathcal{E}}^{\left(\lambda^{p^l}, \mu^{l^l} ; D^{\prime}\right)}, \hat{\mathbb{G}}_m\right) \rightarrow H_0^2\left(\hat{\mathcal{E}}^{(\lambda, \mu ; D)}, \hat{\mathbb{G}}_m\right)
$$
induced by $\psi^{(l)}$ is injective over a $\mathbb{Z}_{(p)}$-algebra under a suitable restriction on $\lambda$ and $\mu$.

Received: May 18, 2024
Revised: July 1, 2024
Accepted: July 8, 2024