200 research outputs found
Formal Fibers of Prime Ideals in Polynomial Rings
Let (R,m) be a Noetherian local domain of dimension n that is essentially
finitely generated over a field and let R^ denote the m-adic completion of R.
Matsumura has shown that n-1 is the maximal height possible for prime ideals of
R^ in the generic formal fiber of R. In this article we prove that every prime
ideal of R^ that is maximal in the generic formal fiber of R has height n-1. We
also present a related result concerning the generic formal fibers of certain
extensions of mixed polynomial-power series rings.Comment: 9 pages to appear in MSRI conference proceeding
Examples of non-Noetherian domains inside power series rings
Let R* be an ideal-adic completion of a Noetherian integral domain R and let
L be a subfield of the total quotient ring of R* such that L contains R. Let A
denote the intersection of L with R*. The integral domain A sometimes inherits
nice properties from R* such as the Noetherian property. For certain fields L
it is possible to approximate A using a localzation B of a nested union of
polynomial rings over R associated to A; if B is Noetherian, then B = A. If B
is not Noetherian, we can sometimes identify the prime ideals of B that are not
finitely generated. We have obtained in this way, for each positive integer s,
a 3-dimensional local unique factorization domain B such that the maximal ideal
of B is 2-generated, B has precisely s prime ideals of height 2, each prime
ideal of B of height 2 is not finitely generated and all the other prime ideals
of B are finitely generated. We examine the map Spec A to Spec B for this
example. We also present a generalization of this example to dimension 4. We
describe a 4-dimensional local non-Noetherian UFD B such that the maximal ideal
of B is 3-generated, there exists precisely one prime ideal Q of B of height 3,
the prime ideal Q is not finitely generated. We consider the question of
whether Q is the only prime ideal of B that is not finitely generated, but have
not answered this question.Comment: 32 pages to appear in JC
Excellent Normal Local Domains and Extensions of Krull Domains
We consider properties of extensions of Krull domains such as flatness that
involve behavior of extensions and contractions of prime ideals. Let (R,m) be
an excellent normal local domain with field of fractions K, let y be a nonzero
element in m, and let R* denote the (y)-adic completion of R. For a finite set
w of elements of yR* that are algebraically independent over R, we construct
two Krull domains: an intersection domain A that is the intersection of R* with
the field of fractions of K[w], and an approximation domain B to A. If R is
countable with dim R at least 2, we prove that there exist sets w as above such
that the extension R[w] to R*[1/y] is flat. In this case B = A is Noetherian,
but may fail to be excellent as we demonstrate with examples. We present
several theorems involving the construction. These theorems yield examples
where B is properly contained in A and A is Noetherian while B is not
Noetherian, and other examples where B = A is not Noetherian.Comment: 24 pages to appear in JPA
Generic fiber rings of mixed power series/polynomial rings
Let K be a field, m and n positive integers, and X = {x_1,...,x_n}, and Y =
{y_1,..., y_m} sets of independent variables over K. Let A be the polynomial
ring K[X] localized at (X). We prove that every prime ideal P in A^ = K[[X]]
that is maximal with respect to P\cap A = (0) has height n-1. We consider the
mixed power series/polynomial rings B := K[[X]][Y]_{(X,Y)} and C :=
K[Y]_{(Y)}[[X]]. For each prime ideal P of B^ = C that is maximal with respect
to either P \cap B = (0) or P \cap C = (0), we prove that P has height n+m-2.
We also prove that each prime ideal P of K[[X, Y]] that is maximal with respect
to P \cap K[[X]] = (0) is of height either m or n+m-2.Comment: 28 page
EXTENSIONS OF LOCAL DOMAINS WITH TRIVIAL GENERIC FIBER
We consider injective local maps from a local domain R to a local domain S such that the generic fiber of the inclusion map R -\u3e S is trivial, that is P R (0) for every nonzero prime ideal P of S. We present several examples of injective local maps involving power series that have or fail to have this property. For an extension R -\u3e S having this property, we give some results on the dimension of S; in some cases we show dim S = 2 and in some cases dim S = 1
INTERMEDIATE RINGS BETWEEN A LOCAL DOMAIN AND ITS COMPLETION, II
We present results connecting flatness of extension rings to the Noetherian property for certain intermediate rings between an excellent normal local domain and its completion. We consider conditions for these rings to have Cohen-Macaulay formal fibers. We also present several examples illustrating these results
IDEALWISE ALGEBRAIC INDEPENDENCE FOR ELEMENTS OF THE COMPLETION OF A LOCAL DOMAIN
Over the past forty years many examples in commutative algebra have been constructed using the following principle: Let k be a field, let S k[xl Xn]x,xn) be a localized polynomial ring over k, and let a be an ideal in the completion S of S such that the associated prims of a are in the generic formal fiber of S; that is, p N S (0) for each p Ass(S/a):. Then S embeds in S/a, the fraction field Q(S) of S embeds in the fraction ring of S/a, and for certain choices of a, the intersection D Q(S) f3 (S/a) is a local Noetherian domain with completion D S/a.
Examples constructed by this method include Nagata’s first examples of nonexcellent rings [N], Ogoma’s celebrated counterexample to Nagata’s catenary conjecture [O1], [O2], examples of Rotthaus and Brodmann [R1], JR2], [BR1], [BR2], and examples of Nishimura and Weston [Ni], [W]. In fact all examples we know of local Noetherian reduced rings which contain and are of finite transcendence degree over a coefficient field may be realized using this principle
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