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.

{{harvtxt|Chevalley|1945}} showed that every local ring of an algebraic variety is analytically unramified.

{{harvtxt|Schmidt|1936}} 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.{{CN|date=March 2022}} This prompted {{harvtxt|Zariski|1948}} to ask whether a local Noetherian domain such that its integral closure is a finite module is always analytically unramified. However {{harvtxt|Nagata|1955}} 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 {{harvs|txt|authorlink=David Rees (mathematician)|first=David|last=Rees|year=1961}} 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 $n_j \to \infty$ 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 {{brackets|x,y}} generated by K and K² {{brackets|x,y}} 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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{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative algebra. Vol. II | origyear=1960 | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90171-8 |mr=0389876 | year=1975}}

Category:Commutative algebra