Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-26T10:56:42.352Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERALISED IMAGINARIES AND GALOIS COHOMOLOGY

Published online by Cambridge University Press:  14 September 2016

DMITRY SUSTRETOV
DMITRY SUSTRETOV*
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS EDMOND J. SAFRA CAMPUS THE HEBREW UNIVERSITY OF JERUSALEM GIVAT RAM, JERUSALEM 91904, ISRAELE-mail: sustretov@ma.huji.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

The objective of this article is to characterise elimination of finite generalised imaginaries as defined in [9] in terms of group cohomology. As an application, I consider series of Zariski geometries constructed [10, 23, 24] by Hrushovski and Zilber and indicate how their nondefinability in algebraically closed fields is connected to eliminability of certain generalised imaginaries.

Keywords

generalised imaginariesinterpretabilityGalois cohomologyinternalityZariski geometriesAzumaya algebras
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 3 , September 2016 , pp. 917 - 935
Copyright
Copyright © The Association for Symbolic Logic 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Auslander, M. and Goldman, O., The Brauer group of a commutative ring . Transactions of the American Mathematical Society, vol. 97 (1960), no. 3, pp. 367–409.Google Scholar
Behrend, K. and Xu, P., Differentiable stacks and gerbes . Journal of Symplectic Geometry, vol. 9 (2011), no 3, pp. 285–341.CrossRefGoogle Scholar
Gille, P. and Szamuely, T., Central Simple Algebras and Galois Cohomology, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2006.Google Scholar
Giraud, J., Cohomologie non-abélienne, Grundlehren der mathematischen Wissenschaften, vol. 179, Springer-Verlag, 1971.CrossRefGoogle Scholar
Goodrick, J., Kim, B., and Kolesnikov, A., Amalgamation functors and boundary properties in simple theories . Israel Journal of Mathematics, vol. 193 (2013), no. 1, pp. 169–207.Google Scholar
Goodrick, J. and Kolesnikov, A., Groupoids, covers, and 3-uniqueness in stable theories, this Journal, vol. 75 (2010), no. 3, pp. 905–929.Google Scholar
Grothendieck, A., Le groupe de Brauer I-III , Dix exposés sur la cohomolgie des schémas, North-Holland, Amsterdam, 1968.Google Scholar
Hrushovski, E., A new strongly minimal set . Annals of Pure and Applied Logic, vol. 62 (1993), no. 2, pp. 147–166.CrossRefGoogle Scholar
Hrushovski, E., Groupoids, imaginaries and internal covers . Turkish Journal of Mathematics, vol. 36 (2012), no. 2, pp. 173–198.Google Scholar
Hrushovski, E. and Zilber, B., Zariski geometries . Journal of the American Mathematical Society, vol. 9 (1996), no. 1, pp. 1–56.Google Scholar
de Jongh, A., A result of Gabber, http://www.math.columbia.edu/∼dejong/papers/2-gabber.pdf, 2004.Google Scholar
Knus, M-A. and Ojanguren, M., Théorie de la descente et algèbres d’azumaya, Springer-Verlag, Berlin/Heidelberg, 1974.Google Scholar
Lomonosov, V. and Rosenthal, P., The simplest proof of Burnside’s theorem on matrix algebras . Linear Algebra and its Applications, vol. 383 (2004), pp. 45–48.Google Scholar
Milne, J., Étale Cohomology, Princeton University Press, Princeton, 1980.Google Scholar
Neukirch, J., Schmidt, A., and Wingberg, K., Cohomology of Number Fields, Grundlehren der mathematischen Wissenschaften, Springer, Berlin/Heidelberg, 2008.Google Scholar
Pillay, A., Remarks on Galois cohomology and definability, this Journal, vol. 62 (1997), no. 2, pp. 487–492.Google Scholar
Pillay, A., Algebraically closed fields and model theory , Model Theory and Algebraic Geometry (Bouscaren, E., editor), Springer-Verlag, Berlin, 1998, pp. 61–84.Google Scholar
Poizat, B., Une théorie de Galois imaginaire, this Journal, vol. 48 (1983), no. 4, pp. 1151–117.Google Scholar
Poizat, B., Stable Groups, Mathematical Surveys and Monographs, no. 87, American Mathematical Society, Providence, RI, 2001.CrossRefGoogle Scholar
Serre, J.-P., Galois Cohomology, Springer-Verlag, Berlin/Heidelberg, 1964.Google Scholar
Zil’ber, B. I., The Structure of Models of Uncountably Categorical Theories, Proceedings of the International Congress of Mathematicians (Warsaw, 1983), vol. 1, 2, PWN, Warsaw, 1984, pp. 359–368.Google Scholar
Zilber, B., Model theory and algebraic geometry , Proceedings of the 10th Easter Conference on Model Theory (Weese, M. and Walter, H., editors), Humboldt Universität, Berlin, 1993, pp. 94720–3480.Google Scholar
Zilber, B., A class of quantum Zariski geometries , Model Theory with Applications to Algebra and Analysis, I and II (Chatzidakis, Z., Macpherson, H. D., Pillay, A., and Wilkie, A. J., editors), Cambridge University Press, Cambridge, 2008, p. 293.Google Scholar
Zilber, B., Non-commutative Zariski geometries and their classical limit . Confluentes Mathematici, vol. 2 (2010), no. 2, pp. 265–291.Google Scholar
Zilber, B., Zariski Geometries: Geometry from the Logician’s Point of View, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2010.Google Scholar
Solanki, V., Sustretov, D., and Zilber, B., The quantum harmonic oscillator as a Zariski geometry , Annals of Pure and Applied Logic, vol. 165 (2014), no 6, pp. 1149–1168.Google Scholar