On valuation spectra

Abstract

If K is an ordered field then every convex subring of K is a valuation ring of K. This easy but fundamental observation has made valuation theory a very natural and important tool in real algebraic geometry. In particular many topological phenomena of semialgebraic sets and of constructible subsets of real spectra are best explained by use of valuations. We have seen in recent years how important it is to switch from the consideration of particular orderings of fields to a study of the set of all orderings of all residue class fields of a commutative ring A, i.e. the real spectrum SperA of A. Now why not do the same with valuations? This leads to the definition of valuation spectra. In principle the points of the valuation spectrum SpevA should be pairs (p, v) consisting of a prime ideal p of A, i.e. a point of SpecA, and a Krull valuation v of the residue class field Quot(A/p). Here one has to made a decision whether one should distinguish between different valuations of Quot(A/p) which have the same valuation ring or not. One further has to choose a topology on SpevA, where again at least two reasonable choices can be made. Finally one should look for sheaves of “functions” on SpevA and some prominent subsets of SpevA. In recent years various authors have defined valuation spectra and/or related spaces. (Brumfiel, de la Puente, Berkovich, Robson, Huber, Schwartz). To my opinion the question which valuation spectrum is the “right” one depends on the applications one has in mind. Certain valuation spectra are important both for real algebraic and for p-adic geometry. In want to stress here a direction followed by Roland Huber which leads to a new foundation of rigid analytic geometry. Huber defines for A in a certain class of topological rings, which he calls “f -adic rings”, a ringed space SpaA, the analytic spectrum of A. The points of SpaA are those points (p, v) of the valuation spectrum SpevA such that a homomorphism form A to a valued field K inducing v is continuous. Analytic spectra are the building blocks of “adic