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
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 OZpW(O)?

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


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

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