Analytically unramified ring

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.

Template:Harvtxt showed that every local ring of an algebraic variety is analytically unramified. Template:Harvtxt 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.Template:CN This prompted Template:Harvtxt to ask whether a local Noetherian domain such that its integral closure is a finite module is always analytically unramified. However Template:Harvtxt 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 Template:Harvs 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 $n_j\to \mathrm{\infty }$ 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({u_n, v_n : n ≥ 0}), where the u_n and v_n are indeterminates. Let T be the subring of the formal power series ring K Template:Brackets generated by K and K² Template:Brackets 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.

References

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