Higher local field

In mathematics, a higher (-dimensional) local field is an important example of a complete discrete valuation field. Such fields are also sometimes called multi-dimensional local fields.

On the usual local fields (typically completions of number fields or the quotient fields of local rings of algebraic curves) there is a unique surjective discrete valuation (of rank 1) associated to a choice of a local parameter of the fields, unless they are archimedean local fields such as the real numbers and complex numbers. Similarly, there is a discrete valuation of rank n on almost all n-dimensional local fields, associated to a choice of n local parameters of the field. In contrast to one-dimensional local fields, higher local fields have a sequence of residue fields. There are different integral structures on higher local fields, depending how many residue fields information one wants to take into account.

Geometrically, higher local fields appear via a process of localization and completion of local rings of higher dimensional schemes. Higher local fields are an important part of the subject of higher dimensional number theory, forming the appropriate collection of objects for local considerations.

Finite fields have dimension 0 and complete discrete valuation fields with finite residue field have dimension one (it is natural to also define archimedean local fields such as R or C to have dimension 1), then we say a complete discrete valuation field has dimension n if its residue field has dimension n−1. Higher local fields are those of dimension greater than one, while one-dimensional local fields are the traditional local fields. We call the residue field of a finite-dimensional higher local field the 'first' residue field, its residue field is then the second residue field, and the pattern continues until we reach a finite field.

Two-dimensional local fields are divided into the following classes:

Higher local fields appear in a variety of contexts. A geometric example is as follows. Given a surface over a finite field of characteristic p, a curve on the surface and a point on the curve, take the local ring at the point. Then, complete this ring, localise it at the curve and complete the resulting ring. Finally, take the quotient field. The result is a two-dimensional local field over a finite field.

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