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On the reduction of semialgebraic sets by real valuations. (English) Zbl 0826.14038
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, 75-95 (1994).
Let $$R$$ be a non-archimedean real closed field. Denote its valuation ring by $$V$$ and its residue field by $$\overline R$$. The paper studies reduction of definable subsets $$S$$ of $$V^n$$, i.e. subsets of $$V^n$$ that are defined by a formula in the first-order language of ordered valuation fields. By the reduction of $$S$$ is meant the image $$\overline S$$ of $$S$$ under the residue map $$V^n \to \overline R^n$$.
The author gives a geometric proof of the (known) fact that $$\overline S$$ is semi-algebraic whenever $$S$$ is definable. He defines a reduction map $$r : H_p (S,T) \to H_p (\overline S, \overline T)$$ on relative homology groups with coefficients in $$\mathbb{Z}/2\mathbb{Z}$$ for a pair of semi- algebraic subsets $$T \subseteq S$$ of $$V^n$$. Functoriality of this reduction map with respect to so-called moderate maps is proved. Finally, the author proves that this reduction map behaves well with respect to the mod-2 intersection product.
For the entire collection see [Zbl 0788.00051].
Reviewer: J.Huisman (Rennes)

##### MSC:
 14P10 Semialgebraic sets and related spaces 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 12J25 Non-Archimedean valued fields 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) 13F30 Valuation rings