Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The étale homotopy type of varieties over $ {\bf R}$


Author: David A. Cox
Journal: Proc. Amer. Math. Soc. 76 (1979), 17-22
MSC: Primary 14F20
MathSciNet review: 534381
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a variety X over $ {\text{Spec}}({\mathbf{R}})$, the étale homotopy type of X is computed in terms of the action of complex conjugation on the complex points $ X({\mathbf{C}})$. This enables one to show that $ X({\mathbf{R}}) \ne \emptyset $ is equivalent to various conditions on the étale cohomology of X, and, when X is a smooth, geometrically connected, proper curve over $ {\text{Spec}}({\mathbf{R}})$, to compute the étale cohomology. Finally, there is a negative result, showing that étale cohomology cannot be used to compute the topological degree of a mapping germ $ f:({{\mathbf{R}}^n},0) \to ({{\mathbf{R}}^n},0)$ .


References [Enhancements On Off] (What's this?)

  • [1] Théorie des topos et cohomologie étale des schémas. Tome 3, Lecture Notes in Mathematics, Vol. 305, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat. MR 0354654
  • [2] M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, vol. 100, Springer-Verlag, Berlin, 1986. Reprint of the 1969 original. MR 883959
  • [3] M. Artin and J. L. Verdier, Seminar on étale cohomology of number fields, Mimeographed notes, Woods Hole, 1964.
  • [4] Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR 0413144
  • [5] David A. Cox, Homotopy theory of simplicial schemes, Compositio Math. 39 (1979), no. 3, 263–296. MR 550644
  • [6] P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, Vol. 569, Springer-Verlag, Berlin-New York, 1977. Séminaire de Géométrie Algébrique du Bois-Marie SGA 41\over2; Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier. MR 0463174
  • [7] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 0217086

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 14F20

Retrieve articles in all journals with MSC: 14F20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0534381-4
Article copyright: © Copyright 1979 American Mathematical Society