Analytically unramified ring explained

In algebra, an analytically unramified ring is a local ring whose completion is reduced (has no nonzero nilpotent).

The following rings are analytically unramified:

• pseudo-geometric reduced ring.
• excellent reduced ring.

showed that every local ring of an algebraic variety is analytically unramified. gave an example of an analytically ramified reduced local ring. Krull showed that every 1-dimensional normal Noetherian local ring is analytically unramified; more precisely he showed that a 1-dimensional normal Noetherian local domain is analytically unramified if and only if its integral closure is a finite module. This prompted to ask whether a local Noetherian domain such that its integral closure is a finite module is always analytically unramified. However gave an example of a 2-dimensional normal analytically ramified Noetherian local ring. Nagata also showed that a slightly stronger version of Zariski's question is correct: if the normalization of every finite extension of a given Noetherian local ring R is a finite module, then R is analytically unramified.

There are two classical theorems of that characterize analytically unramified rings. The first says that a Noetherian local ring (R, m) is analytically unramified if and only if there are a m-primary ideal J and a sequence

n _{j} \to inf ty such that \overline{J ^{j}} \subset J ^{n _{j}} , where the bar means the integral closure of an ideal. The second says that a Noetherian local domain is analytically unramified if and only if, for every finitely-generated R-algebra S lying between R and the field of fractions K of R, the integral closure of S in K is a finitely generated module over S. The second follows from the first.

Nagata's example

Let K₀ be a perfect field of characteristic 2, such as F₂.Let K be K₀, where the u_n and v_n are indeterminates.Let T be the subring of the formal power series ring K  generated by K and K²  and the element Σ(u_nxⁿ+ v_nyⁿ). Nagata proves that T is a normal local noetherian domain whose completion has nonzero nilpotent elements, so T is analytically ramified.