Fernando Galván, José Francisco and Gamboa, J. M. (2012) On the semialgebraic Stone-Čech compactification of a semialgebraic set. Transactions of the American Mathematical Society (364). 3479-3511 . ISSN 1088-6850
Restringido a Repository staff only hasta 31 December 2020.
In the same vein as the classical Stone–ˇCech compactification, we prove in this work that the maximal spectra of the rings of semialgebraic and bounded semialgebraic functions on a semialgebraic set M ⊂ Rn, which are homeomorphic topological spaces, provide the smallest Hausdorff compactification of M such that each bounded R-valued semialgebraic function on M extends continuously to it. Such compactification β∗sM, which can be characterized as the smallest compactification that dominates all semialgebraic compactifications of M, is called the semialgebraic Stone– ˇ Cech compactification of M, although it is very rarely a semialgebraic set. We are also interested in determining the main topological properties of the remainder ∂M = β∗sM \M and we prove that it has finitely many connected components and that this number equals the number of connected components of the remainder of a suitable semialgebraic compactification of M. Moreover, ∂M is locally connected and its local compactness can be characterized just in terms of the topology of M.
|Uncontrolled Keywords:||Semialgebraic function, maximal spectrum, semialgebraic compactification, semialgebraic Stone–Čech compactification, remainder|
|Subjects:||Sciences > Mathematics > Algebraic geometry|
Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998. Translated from the 1987 French original; Revised by the authors.
Nicolas Bourbaki, General topology. Chapters 1–4, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1966 edition.
Hans Delfs and Manfred Knebusch, Separation, retractions and homotopy extension in semialgebraic spaces, Pacific J. Math. 114 (1984), no. 1, 47–71.
J.F. Fernando: On chains of prime ideals in rings of semialgebraic functions. Preprint RAAG (2010). http://www.mat.ucm.es/josefer/pdfs/preprint/chains.pdf
J.F. Fernando: On distinguished points of the remainder of the semialgebraic Stone-Čech compactification of a semialgebraic set. Preprint RAAG (2010). http://www.mat.ucm.es/josefer/pdfs/preprint/remainder.pdf
J.F. Fernando, J.M. Gamboa: On Łojasiewicz's inequality and the Nullstellensatz for rings of semialgebraic functions. Preprint RAAG (2010). http://www.mat.ucm.es/josefer/pdfs/preprint/null-loj.pdf
J.F. Fernando, J.M. Gamboa: On the Krull dimension of rings of semialgebraic functions. Preprint RAAG (2010). http://www.mat.ucm.es/josefer/pdfs/preprint/dim.pdf
J.F. Fernando, J.M. Gamboa: On the spectra of rings of semialgebraic functions. Collect. Math., to appear (2012).
J.F. Fernando, J.M. Gamboa: On Banach-Stone type theorems in the semialgebraic setting. Preprint RAAG (2010). http://www.mat.ucm.es/josefer/pdfs/preprint/homeo.pdf
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960
Giuseppe De Marco and Adalberto Orsatti, Commutative rings in which every prime ideal is contained in a unique maximal ideal, Proc. Amer. Math. Soc. 30 (1971), 459–466.
James R. Munkres, Topology: a first course, Prentice-Hall Inc., Englewood Cliffs, N.J., 1975.
|Deposited On:||11 Sep 2012 08:59|
|Last Modified:||02 Mar 2016 14:37|
Repository Staff Only: item control page