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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.

13J30 Real algebra
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[1] C. Andradas, L. Bröcker, J. Ruiz, Constructible sets in real geometry, Springer 1996 · Zbl 0873.14044
[2] E. Becker, D. Gondard, On the space of real places of a formally real field, Real analytic and algebraic geometry, Walter de Gruyter (1995), 21-46 · Zbl 0869.12002
[3] E. Becker, V. Powers, Sums of powers in rings and the real holomorphy ring, J. reine angew. Math. 480 (1996), 71-103 · Zbl 0922.12003
[4] J. Bochnak, M. Coste, M.-F. Roy, Géométrie algébrique réelle, Ergeb. Math. Springer 1987 · Zbl 0633.14016
[5] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43-60 · Zbl 0184.29401
[6] R. Huber, Bewertungsspektrum und rigide geometrie, Regensburger Math. Schriften 23 1993 · Zbl 0806.13001
[7] R. Huber, M. Knebusch, On valuation spectra, Contemporary Math. 155 (1994), 167-206 · Zbl 0799.13002
[8] M. Knebusch, D. Zhang, Manis valuations and Prüfer extensions I, Springer 2002 · Zbl 1033.13001
[9] T.-Y. Lam, An introduction to real algebra, Rky. Mtn. J. Math. 14 (1984), 767-814 · Zbl 0577.14016
[10] M. Marshall, Spaces of orderings and abstract real spectra, Lecture Notes in Mathematics 1636, Springer 1996 · Zbl 0866.12001
[11] M. Marshall, A real holomorphy ring without the Schmüdgen property, Canad. Math. Bull. 42 (1999), 354-358 · Zbl 0971.12002
[12] M. Marshall, Real reduced multirings and multifields J. Pure and Applied Algebra 205 (2006), 452-468 · Zbl 1089.14009
[13] M.J. de la Puente, Riemann surfaces of a ring and compactifications of semi-algebraic sets, Doctoral Dissertation, Stanford 1988
[14] M.J. de la Puente, Specializations and a local homomorphism theorem for real Riemann surfaces of rings, Pac. J. Math. 176 (1996), 427-442 · Zbl 0868.13004
[15] K. Schmüdgen, The \(K\)-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), 203-206 · Zbl 0744.44008
[16] H. Schülting, On real places of a field and the real holomorphy ring, Comm. Alg. 10 (1982), 1239-1284 · Zbl 0509.14026
[17] M. Schweighofer, Iterated rings of bounded elements and generalizations of Schmüdgen’s Positivstellensatz, J. reine angew. Math. 554 (2003), 19-45 · Zbl 1096.13032
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