In algebra, an analytically unramified ring is a

local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...

whose completion is reduced (has no nonzero

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the class ...

). The following rings are analytically unramified: * pseudo-geometric reduced ring. * excellent reduced ring. showed that every local ring of an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...

is analytically unramified. gave an example of an analytically ramified reduced local ring. Krull showed that every 1-dimensional normal

Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...

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 \infty

such that

\overline \subset J^

, where the bar means the

integral closure of an ideal In algebra, the integral closure of an ideal ''I'' of a commutative ring ''R'', denoted by \overline, is the set of all elements ''r'' in ''R'' that are integral over ''I'': there exist a_i \in I^i such that :r^n + a_1 r^ + \cdots + a_ r + a_n = 0. ...

. 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 In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' is ...

of ''S'' in ''K'' is a finitely generated module over ''S''. The second follows from the first.

Nagata's example

Let ''K''₀ be a perfect field of characteristic 2, such as F₂. Let ''K'' be ''K''₀(), where the ''u''_''n'' and ''v''_''n'' are indeterminates. Let ''T'' be the subring of the

formal power series ring In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...

''K'' generated by ''K'' and ''K''^''2'' and the element Σ(''u''_''n''''x''^''n''+ ''v''_''n''''y''^''n''). Nagata proves that ''T'' is a normal local noetherian domain whose completion has nonzero nilpotent elements, so ''T'' is analytically ramified.

References

* * * * * * *{{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 Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, location=Berlin, New York , isbn=978-0-387-90171-8 , mr=0389876 , year=1975 Commutative algebra