Let $\mathcal{O}$ be the ring of integers in a $p$-adic local field, totally ramified over $\mathbb{Q}_p$. We fix a uniformizer $\pi$ and form the ring of relative Witt vectors $W_{\mathcal{O}}(\mathcal{O})$ with respect to $\pi$. See [Fargues, The Curve, p. 4] for a definition of $W_{\mathcal{O}}$. There is a natural map $$W(\mathcal{O})\longrightarrow W_{\mathcal{O}}(\mathcal{O}),$$ where $W(\mathcal{O})$ denotes the usual Witt vectors of $\mathcal{O}$. The map is determined by the property that it commutes with forming the ghost components. Let us consider its $\mathcal{O}$-linear extension $$\mathcal{O}\otimes_{\mathbb{Z}_p}W(\mathcal{O})\longrightarrow W_{\mathcal{O}}(\mathcal{O}).$$

Let $V_\pi$ denote the Verschiebung on $W_{\mathcal{O}}(\mathcal{O})$. It is known that there exists a unit $\varepsilon$ such that $$V_\pi(\varepsilon) = \pi - [\pi].$$ My Question is now as follows: Does this unit come from the ring $\mathcal{O}\otimes_{\mathbb{Z}_p}W(\mathcal{O})$?

Equivalently: The ghost of $\varepsilon$ is $(1-\pi^{p-1},1-\pi^{p^2-1},\ldots,1-\pi^{p^n-1},\ldots)\in \mathcal{O}^{\mathbb{N}}$. So the question is, whether $(\pi^{p-1},\pi^{p^2-1},\ldots)$ lies in the image of the $\mathcal{O}$-linear extension of the ghost map $$\mathcal{O}\otimes_{\mathbb{Z}_p} W(\mathcal{O})\longrightarrow \mathcal{O}^{\mathbb{N}}.$$

  • 3
    $\begingroup$ Can you give a reference where I can see what the relative Witt vectors are? $\endgroup$
    – Lubin
    Aug 24, 2016 at 18:28


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Browse other questions tagged or ask your own question.