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On the connected components of a global semianalytic set

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Ruiz Sancho, Jesús María (1988) On the connected components of a global semianalytic set. Journal fur die Reine und Angewandte Mathematik, 392 . pp. 137-144. ISSN 0075-4102

Official URL: http://www.degruyter.com/view/j/crll.1988.issue-392/crll.1988.392.137/crll.1988.392.137.xml


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Abstract

Let M be a real analytic manifold and O(M) its ring of global analytic functions. A global semianalytic subset of M is any set Z of the form Z=⋃ i=0 r {x∈M:fi1(x)>0,⋯,fis(x)>0,gi(x)=0}, (1.1), where fij,gi∈O(M). This imitates the definitions of semialgebraic sets and semianalytic germs, and gives rise to the same old basic problems: Can the gi's in (1.1) be omitted if Z is open? Is the closure of Z global semianalytic when Z itself is? And the connected components of Z? In an earlier paper [in Algèbre, 84–95, Univ. Rennes I, Rennes, 1986] we showed that this is possible for the first two questions in case M is compact: our method relied upon the theory of the real spectrum. In this note we deal with the third question and prove Theorem 1.2: Let Z be a global semianalytic subset of a real analytic manifold M. Assume that Z is relatively compact. Then the connected components of Z are global semianalytic subsets of M. For the proof, we use again the real spectrum, plus the solution by Ch. Rotthaus of M. Artin's conjecture on the approximation property of excellent rings [Rotthaus, Invent. Math. 88 (1987), no. 1, 39–63].


Item Type:Article
Uncontrolled Keywords:Connected components; global semianalytic subset of a real analytic manifold
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:20294
Deposited On:06 Mar 2013 15:11
Last Modified:12 Dec 2018 15:13

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