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Real holomorphy rings and the complete real spectrum. (English) Zbl 1209.13026
Let $$A$$ be a commutative ring with $$1$$. The prime spectrum $$\mathrm{Spec}\, A$$ consisting of all prime ideals of $$A$$ is a basic object in algebraic geometry. In real geometry where also inequalities are considered the corresponding notion is the real spectrum $$\mathrm{Sper}\, A$$ of $$A$$. It consists of all pairs $$(\mathfrak{p}, P)$$ where $$\mathfrak{p}$$ is a prime ideal of $$A$$ and $$P$$ is an ordering on $$k(\mathfrak{p})$$ which is the quotient field of $$A/\mathfrak{p}$$. (Recall that an ordering on a field $$K$$ is a subset $$P$$ of $$K$$ such that $$P+P\subset P, PP\subset P, P\cup -P=K$$ and $$P\cap -P=\{0\}$$.) Both $$\mathrm{Spec}\, A$$ and $$\mathrm{Sper}\, A$$ carry a natural topology (the Zariski topology in the first case and the Harrison topology in the second case) that make them a spectral space in the sense of [M. Hochster, Trans. Am. Math. Soc. 142, 43–60 (1969; Zbl 0184.29401)]. Motivated for example by rigid geometry, the valuation spectrum $$\mathrm{Spv}\, A$$ was introduced which consists of all pairs $$(\mathfrak{p},v)$$ where $$\mathfrak{p}$$ is a prime ideal of $$A$$ and $$v$$ is a valuation on $$k(\mathfrak{p})$$ (see for example [R. Huber and M. Knebusch, Contemp. Math. 155, 167–206 (1994; Zbl 0799.13002)]). It carries also a natural topology making it a spectral space.
The authors define in the paper under review the complete real spectrum $$\mathrm{Sper}^c A$$ of $$A$$. It is related to the valuation spectrum in the same way as the real spectrum is related to the prime spectrum. It consists of all triples $$(\mathfrak{p},v,P)$$ where $$\mathfrak{p}$$ is a prime ideal of $$A$$, $$v$$ is a valuation on $$k(\mathfrak{p})$$ and $$P$$ is an ordering on the residue field $$B_v/\mathfrak{m}_v$$ of $$v$$. There are natural maps $$\mathrm{Sper}^c A\to\mathrm{Spec}\, A, \mathrm{Sper}^c A\to\mathrm{Spv}\, A$$ and $$\mathrm{Sper}\, A\to\mathrm{Sper}^c A$$. The authors equip $$\mathrm{Sper}^c A$$ with a natural topology by declaring that the sets $$U(a,b):=\{(\mathfrak{p},v,P)\in\mathrm{Sper}^c A: v(a)=v(b)\neq\infty, \frac{a+\mathfrak{p}}{b+\mathfrak{p}}+ \mathfrak{m}_v>0\mathrm{ at }P\}$$ for $$a,b\in A$$ form a subbasis of open sets. They show that this topology makes the complete real spectrum a spectral space. The specialization relation in this spectral space is investigated.
Special attention is paid to the case when the ring $$A$$ in question is a real holomorphy ring. Recall that the real holomorphy ring $$H_A$$ of $$A$$ is given by $$H_A:=\{a\in A:\exists n\in\mathbb{N}\mathrm{ s.t. }-n\leq a\leq n\mathrm{ on }\mathrm{Sper}\, A\}$$. If $$K$$ is a formally real field then $$\mathrm{Sper}^c K$$ can be naturally identified with $$\mathrm{Sper}\, H_K$$. The authors call the ring $$A$$ a real holomorphy ring if $$H_A=A$$. They obtain strong statements about the natural map $$\mathrm{Sper}\, A\to\mathrm{Sper}^c A$$ in the case of a real holomorphy ring.

##### MSC:
 13J30 Real algebra
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##### References:
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