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Complete ideals defined by sign conditions and the real spectrum of a two-dimensional local ring. (English) Zbl 0849.13014
Let $$(A,m)$$ 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.

##### MSC:
 13H05 Regular local rings 14P05 Real algebraic sets 14A05 Relevant commutative algebra 13A50 Actions of groups on commutative rings; invariant theory
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##### References:
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