Formal fibers and birational extensions

Abstract

Suppose (R, m) is a local Noetherian domain with quotient field K and m-adic completion Ȓ. It is well known that the fibers of the morphism Spec(Ȓ) ₒ Spec(R), i.e., the formal fibers of R, encode important information about the structure of R. Perhaps the most important condition in Grothendieck’s definition of R being excellent is that the formal fibers of R be geometrically regular. Indeed, a local Noetherian ring is excellent provided it is universally catenary and has geometrically regular formal fibers [G, (7.8.3), page 214]. But the structure of the formal fibers of R is often difficult to determine. We are interested here in bringing out the interrelatedness of properties of the generic formal fiber of R with the existence of certain local Noetherian domains C birationally dominating R and having C/mC is a finite R-module.

Formal Fibers and Complete Homomorphic Images

“…is m7?-primary, then p is extended from 7? by [HRS,Lemma 1.15], and hence 7?/(pn7?) is complete, so 7?…”

“…It follows (cf. [HRS,Corollary 1.10]) that R/(anR) is complete and (an7? ).R = a for each ideal a of R with ht(a) > 2.…”

On the completeness of factor rings

“…Therefore, by Theorem 2.7 in [3], there is exactly one analytically irreducible normal local Noetherian domain C that birationally dominates A satisfying α(C) = 0 and satisfying C/mC is a finite A-module. (Here, m denotes the maximal ideal of A.…”

“…In particular, one could show that the following result holds: Let (T, M ) be a complete local domain containing the integers and such that |T /M | ≥ c. Let L be a countable set of nonmaximal incomparable prime ideals of T such that T p is a regular local ring for all p ∈ L. Then there exists an excellent local domain A such that the completion of A is T , the maximal ideals in the generic formal fiber of A is exactly the set L and if I is a nonzero ideal of A, then A/I is complete. Then, by Theorem 2.7 in [3], there is a one-to-one correspondence between elements of L and analytically irreducible normal local Noetherian domains C that birationally dominate A, have α(C) = 0 and satisfy that C/mC is a finite A-module where m denotes the maximal ideal of A.…”

“…For example, in [3], Heinzer, Rotthaus and Sally show that if "enough" prime ideas of A satisfy this condition, then there is a one-to-one correspondence between the prime ideals in that are maximal in the generic formal fiber of A and certain birational extensions of A. (See [3] for details.…”