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The following is taken from notes distributed by Matthew Morrow at the Midwest Number Theory Conference for Graduate Students and Recent PhD’s 2010:

A *two-dimensional local field F* is a complete discrete valuation field whose residue field is a local field.

Some examples of two-dimensional local fields are

(i) For any local field , consider . This is complete under the -adic valuation and has residue field .

(ii) Let be the -adic completion of . is a characteristic zero complete DVR in which is prime, and its residue field is .

Suppose that is a two-dimensional scheme of finite type and let be a closed point and an irreducible curve containing . Let and let be the height one prime in which is the local equation of at . Consider the following sequence of localisations and completions:

It follows from the excellence of that is a radical ideal of the completion . We then localise and complete at and again use excellence to deduce that 0 is a radical ideal in the resulting ring, i.e., is reduced. The total field of fractions is therefore isomorphic to a finite direct sum of fields, and each is a two-dimensional local field.

The two-dimensional adelic philosophy, originally due to A. Parshin, is that we should study via the family of two-dimensional local fields .