Complete ideals defined by sign conditions and the real spectrum of a two-dimensional local ring.

*(English)*Zbl 0849.13014Let \((A,m)\) be a regular local ring and let \(\alpha\) and \(\beta\) be points of the real spectrum of \(A\) centered at \(m\). \(\alpha\) and \(\beta\) may be viewed as total ordering on quotients of \(A\). Associated with \(\alpha\) and \(\beta\), there is the so-called “separating ideal”, \(\langle\alpha,\beta\rangle\subseteq A\), which is generated by all \(a \in A\) such that \(a\) is non-negative with respect to \(\alpha\) and \(-a\) is non-negative with respect to \(\beta\). \(\langle\alpha, \beta \rangle\) is a valuation ideal for the valuation canonically associated with \(\alpha\) (or \(\beta)\), and hence is complete. It is known that a thorough understanding of \( \langle \alpha, \beta \rangle\) would contribute greatly to a solution of the long-standing Pierce-Birkhoff conjecture, but up to now no good techniques for working with it have been known. In the paper under review the authors investigates \(\langle\alpha, \beta \rangle\) by applying quadratic transforms to \((A,m)\) and using Zariski’s theory of complete ideals in two-dimensional regular local rings to analyze how \(\langle \alpha, \beta \rangle\) is affected. Suppose that \((A', m')\) is a quadratic transform of \(A\). Under natural hypotheses, \(\alpha\) and \(\beta\) induce points \(\alpha'\) and \(\beta'\) in the real spectrum of \(A'\) which are centered at \(m'\).

The main result of the paper (theorem 4.7) is a formula which relates \(\langle \alpha', \beta' \rangle \subseteq A'\) with the ideal transform \(\langle \alpha, \beta \rangle'\) of \(\langle \alpha, \beta \rangle\) in \(A'\). It says that if \(A\) is two-dimensional and has real closed residue field, and if \(\langle \alpha, \beta \rangle\) is not the maximal ideal, then \(\langle \alpha', \beta' \rangle = \langle \alpha, \beta \rangle'\). – Applications of this result show that the transformation formula provides an essentially complete understanding of separating ideals in two dimensional regular algebras over real closed fields.

The main result of the paper (theorem 4.7) is a formula which relates \(\langle \alpha', \beta' \rangle \subseteq A'\) with the ideal transform \(\langle \alpha, \beta \rangle'\) of \(\langle \alpha, \beta \rangle\) in \(A'\). It says that if \(A\) is two-dimensional and has real closed residue field, and if \(\langle \alpha, \beta \rangle\) is not the maximal ideal, then \(\langle \alpha', \beta' \rangle = \langle \alpha, \beta \rangle'\). – Applications of this result show that the transformation formula provides an essentially complete understanding of separating ideals in two dimensional regular algebras over real closed fields.

Reviewer: N.I.Osetinski (Moskva)

##### MSC:

13H05 | Regular local rings |

14P05 | Real algebraic sets |

14A05 | Relevant commutative algebra |

13A50 | Actions of groups on commutative rings; invariant theory |

##### Keywords:

real spectrum; total ordering; Pierce-Birkhoff conjecture; quadratic transforms; complete ideals; two-dimensional regular local rings; ideal transform; separating ideals##### References:

[1] | Abhyankar, American Journal of Mathematics 78 pp 321– (1956) |

[2] | and , Géométrie Algébrique Réelle, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 12, Springer-Verlag, Berlin –Heidelberg-New York, 1987 |

[3] | Brown, Rocky Mountain J. Math. 1 pp 633– (1971) |

[4] | Complete Ideals in Two-dimensional Regular Local Rings, Commutative algebra, Proc. Microprogram, MSRI publication no. 15, Springer-Verlag. Berlin –Heidelberg–New York. 1989, 325–338 |

[5] | Complete Ideals in Regular Local Rings, Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, 1987, 203–231 |

[6] | Madden, Arch. Math. 53 pp 565– (1989) |

[7] | and , In preparation |

[8] | and , In preparation. |

[9] | Zariski, Amer. J. Math. 60 pp 151– (1938) |

[10] | and , Commutative Algebra, Vol. II, Van Bostrand 1960 · doi:10.1007/978-3-662-29244-0 |

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