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Real spectrum with Nash structural sheaf. (English) Zbl 1062.14070

Using the definition of Nash structural sheaf over the real spectrum of a commutative ring given by M.-F. Roy [in: Geométrie algébrique réelle et formes quadratiques, Lect. Notes Math. 959, 406–432 (1982; Zbl 0497.14009)] the author shows that the Nash structural sheaf is determined only by the underlying topological space. He gives easy descriptions of a stalk of this Nash sheaf and sections of this sheaf over an open basis. He also calculates the stalks and global sections of it in a restricted case. As an application, the author investigates some basic properties of “separated” morphisms of real schemes.

MSC:

14P20 Nash functions and manifolds
13J30 Real algebra

Citations:

Zbl 0497.14009
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Full Text: DOI Euclid

References:

[1] Alonso, M. E., and Roy, M.-F.: Real strict localisation. Math. Z., 194 (3), 429-441 (1987). · Zbl 0595.20061 · doi:10.1007/BF01162248
[2] Coste, M., Ruiz, M., and Shiota, M.: Uniform bounds on complexity and transfer of global properties of Nash functions. J. Reine Angew. Math., 536 , 209-235 (2001). · Zbl 0981.14028 · doi:10.1515/crll.2001.056
[3] Hartshorne, R.: Algebraic Geometry. Springer-Verlag, New York-Heidelberg (1977). · Zbl 0367.14001
[4] Coste, M., Bochnak, J., and Roy, M.-F.: Real Algebraic Geometry. Translated from the 1987 french original “Géométrie Algébrique Réele”. Springer-Verlag, Berlin (1998).
[5] Matsumura, H.: Commutative Ring Theory. Translated from the Japanese by Reid, M. Cambridge Univ. Press, Cambridge (1986). · Zbl 0603.13001
[6] Raynaud, M.: Anneaux locaux henséliens. Lecture Note in Math. vol. 169, Springer-Verlag, Berlin-New York, pp. 1-129 (1970). · Zbl 0203.05102
[7] Roy, M.-F.: Faisceau Structual sur le spectre réel et functions de Nash. Real Algebraic Geometry and Quadratic Forms. Lecture Note in Math. vol. 959, Springer-Verlag, Berlin-New York, pp. 406-432 (1982).
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