On the integral closure of polynomial rings over a valuation domain

HAGEN KNAF

Mathematisches Institut der Universität Heidelberg Im Neuenheimer Feld 288, 69120 Heidelberg, Germany

Abstract

Let O_{v} be a valuation domain with field of fractions K and corresponding valuation v. The aim of this paper is to present a collection of results concerning the structure and hereditary properties of the integral closure R of the polynomial ring O_{v}[x] in a finite extension F of the field of fractions K(x) of O_{v}[x].

Precisely the problems treated are:

• Provided R|O_{v}[x] is finite, do integral bases for this ring extension exist ?

• Which polynomial rings O_{v}[x] have finite integral closure in any finite extension of K(x) ?

• Let O_{w} be a valuation ring dominating O_{v} such that the extension of the fields of fractions is algebraic. Under which conditions is the tensor product R ⊗_{O_{v}} O_{w} normal ?

A common feature of our treatment of each of these questions is the use of valuation theoretic methods, as introduced and applied to geometric problems in [GMP3] and [G].

Partial answers to the first question are already known: By [Ge-vdP] integral bases for the extension R|O_{v}[x] exist in the case of a valuation ring O_{v} of Krull dimension 1 such that the field K is complete with respect to the valuation v. For general valuation rings B. Green, M. Matignon and F. Pop in [GMP3] have proved an existence theorem of a different type. They showed that in a given function field F of transcendence degree 1 over K, under certain conditions there are many elements x ∈ F transcendental over K such that the integral closure R of O_{v}[x] in F is a finite free O_{v}[x] - module. In the same paper they posed the general question for integral bases of a finite extension R|O_{v}[x].

In section 3 of the present paper a positive answer to this question is given. It is obtained by first applying a theorem of G. Sabbagh [Sab] on the coherence of polynomial rings over valuation rings to show the coherence of the integral closure R itself. Secondly a generalization of the theorem of Quillen and Suslin about projective modules over polynomial rings K[X_{1}, ..., X_{n}] due to Y. Lequain and A. Simis [Le-Si] is used to deduce freeness of the ring extension R|O_{v}[x].

Comparing with the noetherian theory of Nagata rings the second question has a “standard” answer: O_{v}[x] has finite integral closure in any finite extension of its field of fractions K(x) if and only if O_{v} has this property. The latter can be expressed in terms of the value group of O_{v}. The proof of this result is a simple application of a transfer principle due to F.K. Kuhlmann [Kuh].

Finally the third question on the normality of base changes R ⊗_{O_{v}} O_{w} is treated by assuming normality of the ring R ⊗_{O_{v}} L, L the field of fractions of O_{w}, and then giving valuation theoretic conditions on the extensions O_{w}|O_{v} and R|O_{v} which ensure normality of the base change. Such conditions are:

1. Equality of the value groups and residue fields of the valuation rings O_{v} and O_{w}.

2. Unramifiedness of the Gauß valuation v_{x} in F and separability of the residue field extension Lw|Kv of O_{w}|O_{v}.

Note that condition 1 is satisfied for the henselization of a valuation ring.

References

[G ] B. Green, On Curves over Valuation Rings and Morphisms to IP^{1}, J. Number Theory 59(1996) 262-290.

[Ge-vdP ] L. Gerritzen, M. van der Put, Schottky groups and Mumford curves, Lecture Notes in Math. 817, Berlin - Heidelberg - New York 1980.

[GMP3 ] B.W. Green, M. Matignon, F. Pop, On valued function fields III, Reductions of algebraic curves, J. Reine Angew. Math. 432(1992) 117-133.

[Kuh ] F.V. Kuhlmann, Henselian Function Fields and Tame Fields, Thesis, Heidelberg(1989).

[Le-Si ] Y. Lequain, A. Simis, Projektive Modules over R[X_{1}, …, X_{n}], R a Prüfer domain, J. Pure Appl. Algebra 18 (1980) 165-171.

[Sab ] G. Sabbagh, Coherence of Polynomial Rings and Bounds in Polynomial Ideals, J. Algebra 31(1974) 499-507.