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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].

##### MSC:
 13A18 Valuations and their generalizations for commutative rings 14A05 Relevant commutative algebra 13J15 Henselian rings 13B40 Étale and flat extensions; Henselization; Artin approximation
##### Keywords:
Bourbaki valuations; valuation spectra; henselization