Local-Global Principle for rings

Luck Darnière Université d’Angers, France e-mail: Luck.Darniere@univ-angers.fr

November 6, 1998

Let us say that a ring A with fraction field K satisfies the Local-Global principle with respect to its henselisations (LGPH) if and only if:

i. For every non zero a in A, the set of all maximal ideals of A wich do not contain A is quasi-compact (with respect to Zariski’s topology).

ii. Every variety defined over K having a smooth point in the henselisation of each valuation ring centered at a maximal ideal of A, has a smooth point in A.

It appears that this property is first order and provides a general and very flexible framework for the model-theoretic study of rings. Specializing this general study to some particular cases leads to natural notions of PAC, PRC or PpC rings, which applies to various “concrete” examples of rings with fraction fields algebraic over Q or F_{p}[T]. Let us give two exemples of applications, among others:

a. The ring of all absolutely p-adic integers –that is those algebraic integers which belong to Q_{p} and such that all their conjugates over Q still belong to Q_{p}– has a decidable complete theory.

b. A PAC domain A of positive characteristic, that is a domain satisfying LGPH with a PAC fraction field of positive characteristic, has a decidable theory if and only if its fraction field has so . If moreover the theories of the fraction field and the residual lattice of A are model-complete (the residual lattice associated to any ring is an interpretable structure which has been introduced by Prestel and Schmid, it plays a key role for the model theory of rings satisfying LGPH) then so is the theory of the r-domain (A, ≤_{A}), and this last result remains true in zero characteristic.