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THE SHORT EXACT SEQUENCE IN DEFINABLE GALOIS COHOMOLOGY

Part of: Number theory: Connections with logic Homological methods (field theory) Connections with logic

Published online by Cambridge University Press:  30 January 2025

DAVID MERETZKY
DAVID MERETZKY*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME NOTRE DAME, IN 46556 USA
*
E-mail: [email protected]
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Abstract

In [2], Pillay introduced definable Galois cohomology, a model-theoretic generalization of Galois cohomology. Let M be an atomic and strongly $\omega $-homogeneous structure over a set of parameters A. Let B be a normal extension of A in M. We show that a short exact sequence of automorphism groups $1 \to \operatorname {\mathrm {Aut}}(M/B) \to \operatorname {\mathrm {Aut}}(M/A) \to \operatorname {\mathrm {Aut}}(B/A) \to 1$ induces a short exact sequence in definable Galois cohomology. We also discuss compatibilities with [3]. Our result complements the long exact sequence in definable Galois cohomology developed in [4].

Keywords

model theoryGalois cohomology

MSC classification

Primary: 11U09: Model theory 12G05: Galois cohomology 12L12: Model theory
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 15
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

REFERENCES

Kolchin, E. R., Differential Algebraic Groups, Academic Press, Orlando, FL, 1985.Google Scholar
Pillay, A., Remarks on Galois cohomology and definability . Journal of Symbolic Logic, vol. 62 (1997), no. 2, pp. 487–492.Google Scholar
Pillay, A., Automorphism groups of prime models, and invariant measures, preprint, 2024, arXiv:2405.11878 [math.LO].Google Scholar
Sánchez, O. L., Meretzky, D., and Pillay, A., More on Galois cohomology, definability, and differential algebraic groups . The Journal of Symbolic Logic, vol. 89 (2024), no. 2, pp. 496–515.Google Scholar
Serre, J.-P., Galois Cohomology, Lecture Notes in Mathematics, vol. 5, Springer, Berlin, 1979.Google Scholar
Tent, K. and Ziegler, M., A Course in Model Theory, Lecture Notes in Logic, vol. 40, Cambridge University Press, Cambridge, 2012.Google Scholar