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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 $K$, consider $F=K((t))$. This is complete under the $t$-adic valuation and has residue field $K$.

(ii) Let $F$ be the $p$-adic completion of $\mathbb{Z}_p((t))$. $F$ is a characteristic zero complete DVR in which $p$ is prime, and its residue field is $\mathbb{F}_p((t))$.

Suppose that $X$ is a two-dimensional scheme of finite type and let $x\in X$ be a closed point and $y\subset X$ an irreducible curve containing $x$. Let $A=\mathcal{O}_{X,x}$ and let $\mathfrak{p}$ be the height one prime in $A$ which is the local equation of $y$ at $x$. Consider the following sequence of localisations and completions:

$A \rightsquigarrow \hat{A} \rightsquigarrow \hat{A}_{\mathfrak{p}\hat{A}} \rightsquigarrow \widehat{\hat{A}_{\mathfrak{p}\hat{A}}} \rightsquigarrow {\rm Frac}\widehat{\hat{A}_{\mathfrak{p}\hat{A}}}=: F_{x,y}$

It follows from the excellence of $A$ that $\mathfrak{p}\hat{A}$ is a radical ideal of the completion $\hat{A}$. We then localise and complete at $\mathfrak{p}\hat{A}$ and again use excellence to deduce that 0 is a radical ideal in the resulting ring, i.e., $\widehat{\hat{A}_{\mathfrak{p}\hat{A}}}$ is reduced. The total field of fractions $F_{x,y}$ 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 $X$ via the family of two-dimensional local fields $(F_{x,y})_{x\in y\subset X}$.