Let O be the ring of integers in a p-adic local field, totally ramified over Qp. We fix a uniformizer π and form the ring of relative Witt vectors WO(O) with respect to π. See [Fargues,
The Curve, p. 4] for a definition of WO.
There is a natural map
where W(O) denotes the usual Witt vectors of O. The map is determined by the property that it commutes with forming the ghost components. Let us consider its O-linear extension
Let Vπ denote the Verschiebung on WO(O). It is known that there exists a unit ε such that
My Question is now as follows: Does this unit come from the ring O⊗ZpW(O)?
The ghost of ε is (1−πp−1,1−πp2−1,…,1−πpn−1,…)∈ON. So the question is, whether (πp−1,πp2−1,…) lies in the image of the O-linear extension of the ghost map