Real holomorphy rings and the complete real spectrum
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 [\textit{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 [\textit{R. Huber} and \textit{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.
scientific article; zbMATH DE number 5799081