On valuation spectra. (English) Zbl 0799.13002

Jacob, William B. (ed.) et al., Recent advances in real algebraic geometry and quadratic forms. Proceedings of the RAGSQUAD year, Berkeley, CA, USA, 1990-1991. Providence, RI: American Mathematical Society. Contemp. Math. 155, 167-206 (1994).
This is a comprehensive account of valuation spectra as independently defined by R. Huber in “Bewertungsspektrum und rigide Geometrie” (Habilitationsschrift, Universität Regensburg 1990), and by M. J. de la Puente in a thesis (Stanford University 1988). For a commutative ring \(A\) with 1, the valuation spectrum \(\text{Spv} A\) is the set \(S(A)\) of equivalence classes of Bourbaki valuations of \(A\) equipped with the topology \({\mathcal T}\) generated by the sets of form \[ \{v \in S (A) \mid v(a) \geq v(b) \neq \infty\} \] with \(a,b \in A\). For \(A\) a field, \(\text{Spv} A\) is effectively Zariski’s abstract Riemann surface. The usefulness of valuation spectra in algebraic geometry is briefly demonstrated in chapter 4.
In chapter 1, the specialization of a valuation in the spectral space \(\text{Spv} A\) is analysed. Topics in chapter 2 include: in section (2.1), for a ring homomorphism \(f:A \to B\), the relation between specializations of \(v\) in \(\text{Spv} B\) and specializations of \(\text{Spv} (f)(v)\) in \(\text{Spv} A\); in (2.4), the connected components of pro-constructible subsets of \(\text{Spv} A\). Another topology \({\mathcal T}''\) features throughout the work. \({\mathcal T}''\) is generated by \({\mathcal T}\) and the sets \(\{v \in S(A) \mid v(a) > v(b)\}\). – Properties of \((S(A), {\mathcal T}'')\) include: (2.2) if \(\text{Spec} A\) is Noetherian, the closure of a constructible subset is constructible; (2.3) if \(A\) is universally catenary, a curve selection lemma holds.
Let \(\alpha\) be a valuation on a field \(k\), and let \(A\) be a \(k\)-algebra. Chapter 3 concerns the subspace \(\text{Spv} (\alpha,A)\) of all \(v\) in \(\text{Spv} A\) such that \(v | k = \alpha\). Its connected components correspond to those of \(\text{Spec} A \otimes_ k K\) where \(K\) is a henselization of \((k,\alpha)\). Further topics include, in (3.1.5), the combinatorial dimension of \(\text{Spv} (\alpha,A)\) and, in (3.3), the use of filters of a certain set of discs in \(k\) to study the example \(\text{Spv} (\alpha,k[T])\) with \(T\) an indeterminate.
For the entire collection see [Zbl 0788.00051].


13A18 Valuations and their generalizations for commutative rings
14A05 Relevant commutative algebra
13J15 Henselian rings
13B40 Étale and flat extensions; Henselization; Artin approximation