Christel Rotthaus

Formal fibers and birational extensions

Suppose (i?, m) is a local Noetherian domain with quotient field K and m-adic completion R . It is well known that the fibers of the morphism Spec(i?) —* Spec(R), i.e., the formal fibers of R, encode important information about the structure of R. Perhaps the most important condition in Grothendiecks 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 / m C is a finite i?-module. The possible dimensions of the formal fibers of a local Noetherian ring are considered in [Ma2] and in [R3]. Following Matsumura in [Ma2], we use a(A) to denote the maximal dimension of a formal fiber of a local Noetherian ring A If (/?, m) is a local Noetherian domain with quotient field K and m-adic completion R, then Matsumura shows [Ma2, Corollary 1, page 262] that a(R) is the dimension of the generic formal fiber R[K\. He also observes that if R is of positive dimension, then a(R) r > 1 are integers, and xu... >xn are indeterminates, then two interesting examples considered in [Ma2] are the rings

On the approximation property of excellent rings

where D is a smooth finite type A-algebra? A theorem, which would imply the existence of an embedding of the above type, yields the positive solution of the Bass-Quillen conjecture. So far this theorem is only known in special cases. There is the following result by Artin [3, 4].

On rings with low dimensional formal fibres

can possibly be 0, IE - 2 and IZ - 1. Moreover, he asked if any number in between 0 and n - 2 can occur as dimension of the generic formal fibre of some local noetherian ring of dimension 12. The purpose of this paper is to construct for any pair of numbers n,t with 0 <

Mixed polynomial/power series rings and relatinos among their spectra

C ↪→ D1 := k[x] [[y/x]] ↪→ · · · ↪→ Dn := k[x] [[y/x]] ↪→ · · · ↪→ E. (2) With regard to Equation 2, for n a positive integer, the map C ↪→ Dn is not flat, but Dn ↪→ E is a localization followed by an adic completion of a Noetherian ring and therefore is flat. We discuss the spectra of these rings and consider the maps induced on the spectra by the inclusion maps on the rings. For example, we determine whether there exist nonzero primes of one of the larger rings that intersect a smaller ring in zero. We were led to consider these rings by questions that came up in two contexts. The first motivation is from the introduction to the paper [AJL] by AlonzoTarrio, Jeremias-Lopez and Lipman: If a map between Noetherian formal schemes can be factored as a closed immersion followed by an open one, can this map also be factored as an open immersion followed by a closed one? This is not true in general. As mentioned in [AJL], Brian Conrad observed that a counterexample can be constructed for every triple (R, x, p), where

Formal fibers and complete homomorphic images

Let (R, m) be an excellent normal local Henselian domain, and suppose that q is a prime ideal in R of height > 1 . We show that, if R/q is not complete, then there are infinitely many height one prime ideals p C qi? of R with pCiR = 0 ; in particular, the dimension of the generic formal fiber of R is at least one. This result may in fact indicate that a much stronger relationship between maximal ideals in the formal fibers of an excellent Henselian local ring and its complete homomorphic images is possibly satisfied. The second half of the paper is concerned with a property of excellent normal local Henselian domains R with zero-dimensional formal fibers. We show that for such an R one has the following good property with respect to intersection: for any field L such that @(R) CiC S(R), the ring LnR is a local Noetherian domain which has completion R.

Open loci of graded modules

Let A = ⊕ i∈N A i be an excellent homogeneous Noetherian graded ring and let M = ⊕ n∈Z M n be a finitely generated graded A-module. We consider M as a module over A 0 and show that the (S k )-loci of M are open in Spec(A 0 ). In particular, the Cohen-Macaulay locus U 0 CM = {p ∈ Spec(A 0 ) | M p is Cohen-Macaulay} is an open subset of Spec(A 0 ). We also show that the (S k )-loci on the homogeneous parts M n of M are eventually stable. As an application we obtain that for a finitely generated Cohen-Macaulay module M over an excellent ring A and for an ideal I C A which is not contained in any minimal prime of M, the (S k )-loci for the modules M/I n M are eventually stable.

Descent of the canonical module in rings with the approximation property

Let (R,m) be a local Noetherian Cohen-Macaulay ring with the approximation property. We show that R admits a canonical module.

Approximating discrete valuation rings by regular local rings

Let k be a field of characteristic zero and let (V, n) be a discrete rank-one valuation domain containing k with V/n = k. Assume that the fraction field L of V has finite transcendence degree s over k. For every positive integer d < s, we prove that V can be realized as a directed union of regular local k-subalgebras of V of dimension d.

On a class of coherent regular rings

The paper investigates a special class of quasi-local rings. It is shown that these rings are coherent and regular in the sense that every finitely generated submodule of a free module has a finite free resolution.

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.