Skip to main content Accessibility help
×
Home

Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup

  • Alessandro Berarducci (a1)

Abstract

By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.

Copyright

References

Hide All
[1]Berarducci, A., O-minimal spectra, infinitesimal subgroups and cohomology, this Journal, vol. 72, (2007), no. 4, pp. 11771193.
[2]Berarducci, A. and Fornasiero, A., O-mininal cohomology: finiteness and invariance results, eprint arXiv:math. LO/0705.3425, 26 May 2007, 28 pp.
[3]Berarducci, A. and Otero, M., An additive measure in o-minimal expansions of fields, Quarterly Journal of Mathematics, vol. 55 (2004), pp. 411419.
[4]Berarducci, A., Otero, M., Peterzil, Y., and Pillay, A., A descending chain condition for groups definable in o-minimal structures, Annals of Pure and Applied Logic, vol. 134 (2005), pp. 303313.
[5]Bredon, G. E., Sheaf theory, second ed., Graduate Texts in Mathematics, no. 170, Springer-Verlag, New York, 1997.
[6]Carral, M. and Coste, M., Normal spectral spaces and their dimension, Journal of Pure and Applied Algebra, vol. 30 (1983), pp. 227235.
[7]Delfs, H., The homotopy axiom in semialgebraic cohomology, Journal für die reine und angewandte Mathematik, vol. 355 (1985), pp. 108128.
[8]Dolich, A., Forking and independence in o-minimal theories, this Journal, vol. 69 (2004), no. 1, pp. 215240.
[9]Edmundo, M., Jones, G. O., and Peatfield, N. J., Sheaf cohomology in o-minimal structures, Journal of Mathematical Logic, vol. 6 (2006), no. 2, pp. 163179.
[10]Edmundo, M., Hurewicz theorems for definable groups, Lie groups and their cohomologies, preprint, 10 13 2007, (http://www.ciul.ul.pt/~edmundo/), 22 pp.
[11]Edmundo, M. and Otero, M., Definably compact abelian groups, Journal of Mathematical Logic, vol. 4 (2004), no. 2, pp. 163180.
[12]Hrushovski, E., Peterzil, Y., and Pillay, A., Groups, measures and the NIP, Journal of the American Mathematical Society, vol. 21 (2008), no. 2, pp. 563596.
[13]Hrushovski, E. and Pillay, A., On NIP and invariant measures, eprint arXiv:math.LO/0710.2330, 11 10 2007, 61 pp.
[14]Jones, G. O., Local to global methods in o-minimal expansions of fields, Ph.D. thesis, Oxford, 2006.
[15]Otero, M. and Peterzil, Y., G-linear sets and torsion points in definably compact groups, eprint arXiv:math.L0/0708.0532vl, 3 08 2007, 17 pp.
[16]Peterzil, Y. and Pillay, A., Generic sets in definably compact groups, Fundamenta Mathematicae, vol. 193 (2007), pp. 153170.
[17]Peterzil, Y., Pillay, A., and Starchenko, S., Definably simple groups in o-minimal structures, Transactions of the American Mathematical Society, vol. 352 (2000), pp. 43974419.
[18]Pillay, A., On groups and rings definable in o-minimal structures, Journal of Pure and Applied Algebra, vol. 53 (1988), pp. 239255.
[19]Pillay, A., Sheaves of continuous definable functions, this Journal, vol. 53 (1988), no. 4, pp. 11651169.
[20]Pillay, A., Type-definability, compact Lie groups and o-minimality, Journal of Mathematical Logic, vol. 4 (2004), pp. 147162.
[21]Shelah, S., Minimal bounded index subgroup for dependent theories, Proceedings of the American Mathematical Society, vol. 136 (2008), pp. 10871091.

Keywords

Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup

  • Alessandro Berarducci (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed