Specific unit in ring of Witt vectors

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
W(O)WO(O),
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
OZpW(O)WO(O).

Let Vπ denote the Verschiebung on WO(O). It is known that there exists a unit ε such that
Vπ(ε)=π[π].
My Question is now as follows: Does this unit come from the ring OZpW(O)?

Equivalently:
The ghost of ε is (1πp1,1πp21,,1πpn1,)ON. So the question is, whether (πp1,πp21,) lies in the image of the O-linear extension of the ghost map
OZpW(O)ON.

Answer

Attribution
Source : Link , Question Author : AndreasK , Answer Author : Community

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