The \(p\)-real spectrum of a commutative ring. (English) Zbl 0782.14046
Let \(K\) be a field. \(K\) is called formally real if \(-1\) is not a sum of squares in \(K\). A theorem of Artin and Schreier asserts that \(K\) is formally real if and only if it can be ordered. Moreover, \(K\) is real closed if and only if it is formally real and no proper algebraic extension of \(K\) is formally real. – Let now \(p\) be a fixed prime number. Denote, for \(n\) natural number, by \(\sum K^{2n}\) the additive semigroup of \(K\) generated by \(x^{2n}\), \(x \in K\). \(K\) is called \(p\)- real if it is formal real and \(\sum K^ 2 \neq \sum K^{2p}\). There exists a characterization of \(p\)-real fields, similar to the theorem of Artin-Schreier, in terms of “orderings of higher level” [see E. Becker, J. Reine Angew. Math. 307/308, 8-30 (1979; Zbl 0398.12012)]. A \(p\)-real field \(K\) is called \(p\)-real closed if \(\sum K^ 2\subseteq\sum L^{2p}\) for every proper algebraic extension \(L\) of \(K\).
The aim of this paper is to construct, for every commutative ring \(A\), a spectral space (i.e., a topological space homeomorphic to the Zariski- spectrum of some commutative ring) \(p\text{-Sper}A\), called the \(p\)-real spectrum of \(A\), which is related to the class of \(p\)-real closed fields in the same manner as the Zariski spectrum \(\text{Spec}A\) (resp., the real spectrum \(\text{Sper}A)\) to the class of algebraically (resp., real) closed fields. The author also studies the topology of \(p\)-Sper\(A\) and the connection of \(p\)-Sper\(A\) with Spec\(A\) and Sper\(A\). In a final section, he considers a morphism of finite presentation \(f:A\to B\), the corresponding map \(f_*:p\text{-Sper}B \to p\text{-Sper}A\) and characterizes the images of the constructible subsets of \(p\text{-Sper}B\) under this map. In contrast to the case of the real spectrum, these images need no longer be constructible.
The proofs are based on some model theoretical results, contained in the 2nd section of the paper, where the author shows that the theory of \(p\)- real closed fields is complete, decidable and admits elimination of quantifiers in an appropriate language.
The aim of this paper is to construct, for every commutative ring \(A\), a spectral space (i.e., a topological space homeomorphic to the Zariski- spectrum of some commutative ring) \(p\text{-Sper}A\), called the \(p\)-real spectrum of \(A\), which is related to the class of \(p\)-real closed fields in the same manner as the Zariski spectrum \(\text{Spec}A\) (resp., the real spectrum \(\text{Sper}A)\) to the class of algebraically (resp., real) closed fields. The author also studies the topology of \(p\)-Sper\(A\) and the connection of \(p\)-Sper\(A\) with Spec\(A\) and Sper\(A\). In a final section, he considers a morphism of finite presentation \(f:A\to B\), the corresponding map \(f_*:p\text{-Sper}B \to p\text{-Sper}A\) and characterizes the images of the constructible subsets of \(p\text{-Sper}B\) under this map. In contrast to the case of the real spectrum, these images need no longer be constructible.
The proofs are based on some model theoretical results, contained in the 2nd section of the paper, where the author shows that the theory of \(p\)- real closed fields is complete, decidable and admits elimination of quantifiers in an appropriate language.
Reviewer: I.Coandă (Bucureşti)
MSC:
| 14P99 | Real algebraic and real-analytic geometry |
| 12D15 | Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) |
| 12J15 | Ordered fields |
| 12L12 | Model theory of fields |
| 03C60 | Model-theoretic algebra |
Citations:
Zbl 0398.12012References:
| [1] | Barton S.M., The real spectrum of higher level of a commutative ring (1988) |
| [2] | Becker E., Lecture Notes,Instituto de Mathematica Pura e Aplicada (1978) |
| [3] | DOI: 10.1515/crll.1979.307-308.8 · Zbl 0398.12012 · doi:10.1515/crll.1979.307-308.8 |
| [4] | DOI: 10.1090/S0273-0979-1986-15431-5 · Zbl 0634.14016 · doi:10.1090/S0273-0979-1986-15431-5 |
| [5] | Becker E., J. reine angew.math 330 pp 53– (1982) |
| [6] | Berr R., Technische Berichte der Fakultãt fÜr Mathematik und Informatik der Universitãt Passau |
| [7] | Berr R., in Math. Z |
| [8] | Bochnak J., Er-gebnisse der Mathematik und ihrer Grenzgebiete |
| [9] | DOI: 10.1016/0022-4049(83)90058-0 · Zbl 0525.14015 · doi:10.1016/0022-4049(83)90058-0 |
| [10] | Coste M., in Contemporary Mathematics 8 pp 27– (1982) |
| [11] | Delon F., Univérsite Paris 7 pp 79– (1990) |
| [12] | DOI: 10.2307/2274582 · Zbl 0681.03017 · doi:10.2307/2274582 |
| [13] | Dries L.v.d., Indag. Math 44 pp 397– (1982) |
| [14] | Endler O., Universitext (1972) |
| [15] | Harman J., Contemporary Mathema\(not;tics 8 pp 141-- (1982)\) |
| [16] | DOI: 10.1090/S0002-9947-1969-0251026-X · doi:10.1090/S0002-9947-1969-0251026-X |
| [17] | Jacob B., Pacific J. of Mathematics 93 pp 95– (1981) |
| [18] | DOI: 10.1515/crll.1981.323.213 · Zbl 0446.12021 · doi:10.1515/crll.1981.323.213 |
| [19] | Knebusch M., J. reine angew. Math 286 pp 314– (1967) |
| [20] | Prieb-Crainpe S., Ergebnisse der Mathematik und ihrer Grenzgebiete 98 |
| [21] | DOI: 10.1090/S0002-9947-1960-0114855-0 · doi:10.1090/S0002-9947-1960-0114855-0 |
| [22] | Sacks G.E., Inc., Reading, Mass (1972) |
| [23] | DOI: 10.1515/crll.1984.347.1 · Zbl 0531.12017 · doi:10.1515/crll.1984.347.1 |
| [24] | Schwartz N., Memoirs of the Am. Math. Soc 397 (1989) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
