Showing changes from revision #3 to #4:
Added | Removed | Changed
In an expansion of a -adic number the are called digits. Usually these digits are defined to be taken elements of the set .
Equivalently In the an digits can be defined to be taken from the setexpansion . Elements of from a this set are calledTeichmüller digits -adic or numberTeichmüller representatives the are called digits. Usually these digits are defined to be taken elements of the set .
The Equivalently the digits can be defined to be taken from the set . is Elements in from bijection this with set the are calledfinite field?Teichmüller digits orTeichmüller representatives . The set of (countably) infinite sequences of elements in hence is in bijection to the set of -adic integers. There is a ring structure on called Witt ring structure such that all ‘’truncated expansion polynomials’‘ called Witt polynomials are morphisms
The set is in bijection with the finite field? . The set of (countably) infinite sequences of elements in hence is in bijection to the set of -adic integers. There is a ring structure on called Witt ring structure such that all ‘’truncated expansion polynomials’‘ called Witt polynomials are morphisms
of groups.
The construction of Witt vectors gives a functorial way to lift a commutative ring of prime characteristic to a commutative ring of characteristic 0. Since this construction is functorial, it can be applied to the structure sheaf of an algebraic variety. In interesting special cases the resulting ring has even more desirable properties: If is perfect is a discrete valuation ring. This is mainly due to the fact that the construction of involves a ring of power series and a ring of power series over a field is always a discrete valuation ring.
nice properties: If is a field is a discrete valuation ring