The Nullstellensatz for real coherent analytic surfaces. (English) Zbl 1215.14058

Let \(X\) be a real coherent analytic space, \({\mathcal O}_X\) the sheaf of analytic functions, \({\mathcal O}(X)\) the ring of global analytic functions. Given an ideal \({\mathfrak a}\subseteq{\mathcal O}(X)\), the zero set of \({\mathfrak a}\) is denoted by \({\mathcal Z}({\mathfrak a})\), the vanishing ideal of \({\mathcal Z}({\mathfrak a})\) is denoted by \({\mathcal I}({\mathcal Z}({\mathfrak a}))\). Nullstellensätze describe the connections between the ideals \({\mathfrak a}\) and \({\mathcal I}({\mathcal Z}({\mathfrak a}))\): When is it true that \({\mathfrak a}={\mathcal I}({\mathcal Z}({\mathfrak a}))\)?
The authors show that, given a real coherent analytic surface \(X\), the equality \({\mathfrak a}={\mathcal I}({\mathcal Z}({\mathfrak a}))\) holds if and only if \({\mathfrak a}\) is a real ideal and is saturated (i.e., \({\mathfrak a}\) is the ideal of global sections of the ideal sheaf generated by \({\mathfrak a}\)). It remains an open question whether the result can be extended to the space \(\mathbb{R}^3\). However, it is shown that the result fails for \(\mathbb{R}^3\) if and only if there is a so-called special irreducible functions that generates a real ideal. Primary ideals and primary decompositions of saturated ideals are among the main tools of the paper.


14P15 Real-analytic and semi-analytic sets
14P99 Real algebraic and real-analytic geometry
32B10 Germs of analytic sets, local parametrization
11E25 Sums of squares and representations by other particular quadratic forms
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