Real spectrum of ring of continuous semialgebraic functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:52:43Z http://mathoverflow.net/feeds/question/9356 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/9356/real-spectrum-of-ring-of-continuous-semialgebraic-functions Real spectrum of ring of continuous semialgebraic functions J Williams 2009-12-19T07:36:15Z 2009-12-19T09:44:48Z <p>Let R be a real closed field, and let U be a semialgebraic subset of $R^n$. Let $S^0(U)$ be the ring of continuous R-valued semialgebraic functions. Also let $\tilde{U}$ be the subset of Spec$_r (R[X_1, \ldots, X_n])$ corresponding to U.</p> <p>What does the real spectrum of $S^0(U)$ look like? Is it related to $\tilde{U}$ in some way? </p> http://mathoverflow.net/questions/9356/real-spectrum-of-ring-of-continuous-semialgebraic-functions/9360#9360 Answer by Jose Capco for Real spectrum of ring of continuous semialgebraic functions Jose Capco 2009-12-19T08:45:30Z 2009-12-19T08:58:51Z <p>The real spectrum of ring of continuous functions (not necessarily semialgebraic) looks exactly like its prime spectrum (i.e. they are homeomorphic), this is also the same for ring of continuous semialgebraic functions (see "<a href="http://books.google.com/books?id=rJk0yn2TROYC&amp;dq=semialgebraic+function+rings&amp;printsec=frontcover&amp;source=bl&amp;ots=ci-7yimvD5&amp;sig=CnSi2RCoOuAMnhcSm6c1JCOOOZ4&amp;hl=en&amp;ei=IZQsS67sMJiXkQWq8LWECQ&amp;sa=X&amp;oi=book%5Fresult&amp;ct=result&amp;resnum=3&amp;ved=0CA0Q6AEwAg" rel="nofollow">Semi-algebraic Function Rings via Reflectors of Partially Ordered Rings</a>" by Schwartz and Madden, Corollary 7.8 ). And $U$ should be homeomorphic to $\text{Spec}_r (S^0(U))$ when we consider their constructible topology (see the book of Schwartz and Madden, Proposition 7.9).</p> http://mathoverflow.net/questions/9356/real-spectrum-of-ring-of-continuous-semialgebraic-functions/9363#9363 Answer by Michel Coste for Real spectrum of ring of continuous semialgebraic functions Michel Coste 2009-12-19T09:44:48Z 2009-12-19T09:44:48Z <p>I don't agree with the preceding answer.</p> <p>When $U$ is a locally compact semialgebraic set, then $\widetilde{U}$ equipped with its sheaf of semi-algebraic continuous functions is isomorphic to the affine scheme $\mathrm{Spec}(S^0(U))$. This is proposition 6 in Carral, Coste : Normal spectral spaces and their dimensions, J. Pure Appl. Algebra 30 (1983) 227-235. In particular $\widetilde{U}$ is homeomorphic to the prime spectrum of $S^0(U)$, which is homeomorphic to its real spectrum. In case $U$ is not locally compact, the situation is different; there are more points in $\mathrm{Spec}(S^0(U))$.</p>