Abstract
We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the finiteness of the genus of simple algebraic groups of type G2. These applications involve the double cosets of adele groups of algebraic groups over arbitrary finitely generated fields: while over number fields these double cosets are associated with the class numbers of algebraic groups and hence have been actively analyzed, similar questions over more general fields seem to come up for the first time. In the Appendix, we link thedoublecosets with Čech cohomology and indicate connections between certain finiteness properties involving double cosets (Condition (T)) and Bass’s finiteness conjecture in K-theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Bass, K-theory and stable algebra, Institut des Hautes Études Scientifiques. Publications Mathématiques 22 (1964) 5–60.
H. Bass, Some problems in “classical” algebraic K-theory, in Algebraic K-Theory, II: Classical Algebraic K-Theory and Connections with Arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Mathematics, Vol. 342, Springer, Berlin-Heidelberg, 1973, pp. 3–73.
A. Borel, Some finiteness properties of adele groups over number fields, Institut des Hautes Etudes Scientifiques. Publications Mathématiques 16 (1963) 5–30.
N. Bourbaki, Commutative Algebra. Chapters 1–7, Elements of Mathematics, Springer, Berlin, 1989.
J.W.S. Cassels and A. Fröhlich (eds.), Algebraic Number Theory, London Mathematical Society, London, 2010.
V. I. Chernousov and V. I. Guletskiı, 2-torsion of the Brauer group of an elliptic curve: generators and relations, Documenta Mathematica Extra Volume (2001), 85–120.
V. I. Chernousov, A. S. Rapinchuk and I. A. Rapinchuk, The genus of a division algebra and the unramified Brauer group, Bulletin of Mathematical Sciences 3 (2013) 211–240.
V. I. Chernousov, A. S. Rapinchuk and I. A. Rapinchuk, Division algebras with the same maximal subfields, Russian Mathematical Surveys 70 (2015) 83–112.
V. I. Chernousov, A. S. Rapinchuk and I. A. Rapinchuk, On the size of the genus of a division algebra, Trudy Matematicheskogo Instituta Imeni V. A. Steklova 292 (2016) 69–99.
V. I. Chernousov, A. S. Rapinchuk and I. A. Rapinchuk, On some finiteness properties of algebraic groups over finitely generated fields, Comptes Rendus Mathématique. Académie des Sciences. Paris 354 (2016) 869–873.
V. I. Chernousov, A. S. Rapinchuk and I. A. Rapinchuk, Spinor groups with good reduction, Compositio Mathematica 155 (2019) 484–527.
B. Farb and R. K. Dennis, Noncommutative Algebra, Graduate Texts in Mathematics, Vol. 144, Springer, New York, 1993.
P. Gille and T. Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge Studies in Advanced Mathematics, Vol. 101, Cambridge University Press, Cambridge, 2006.
G. Harder, Halbeinfache Gruppenschemata über Dedekindringen, Inventiones Mathematicae 4 (1967) 165–191.
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer, New York-Heidelberg, 1977.
B. Kahn, Sur le groupe des classes d’un schema arithmetique, Bulletin de la Société Mathématique de France 134 (2006) 395–415.
T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, Vol. 189, Springer, New York, 1999.
A. S. Merkurjev and A. A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 46 (1982) 1011–1046.
J. S. Meyer, Division algebras with infinite genus, Bulletin of the London Mathematical Society 46 (2014) 463–468.
J. S. Milne, Lectures on Etale Cohomology, available at http://www.jmilne.org/math/CourseNotes/lec.html.
Y. Ouyang, Introduction to Iwasawa Theory, available at https://www.math.unipd.it/~algant/iwasawa.pdf.
V. P. Platonov and A. S. Rapinchuk, Algebraic Groups and Number Theory, Pure and Applied Mathematics, Vol. 139, Academic Press, Boston, MA, 1994.
A. S. Rapinchuk, Strong approximation for algebraic groups, in Thin Groups and Super-strong Approximation, Mathematical Sciences Research Institute Publications, Vol. 61, Cambridge University Press, Cambridge, 2014, pp. 269–298.
A. S. Rapinchuk and I. A. Rapinchuk, On division algebras having the same maximal subfields, Manuscripta Mathematica 132 (2010) 273–293.
A. S. Rapinchuk and I. A. Rapinchuk, Some finiteness results for algebraic groups and unramified cohomology over higher-dimensional fields, https://arxiv.org/abs/2002.01520.
J. Rohlfs, Arithmetische definierte Gruppen mit Galois-operation, Inventiones Mathematicae 4 (1978) 185–205.
L. H. Rowen, Ring Theory. Vols. I, II, Pure and Applied Mathematics, Vols. 127, 128, Academic Press, Boston, MA, 1988.
D. J. Saltman, Lectures on Division Algebras, CMBS Regional Conference Series in Mathematics, Vol. 94, American Mathematical Society, Providence, RI, 1999.
P. Samuel, Anneaux gradués factoriels et modules réflexifs, Bulletin de la Société Mathématique de France 92 (1964) 237–249.
P. Samuel, À propos du théorème des unités, Bulletin des Sciences Mathématiques 90 (1966) 89–96.
J.-P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle, in Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Secrétariat mathématique, Paris, 1958, Exposé 23.
J.-P. Serre, Local Fields, Graduate Texts in Mathematics, Vol. 67, Springer, New York-Berlin, 1979.
J.-P. Serre, Local Algebra, Springer Monographs in Mathematics, Springer, Berlin, 2000.
J.-P. Serre, Galois Cohomology, Springer Monographs in Mathematics, Springer, Berlin, 2002.
R. Sharifi, Iwasawa theory, available at http://math.ucla.edu/~sharifi/iwasawa.pdf.
S. V. Tikhonov, Division algebras of prime degree with infinite genus, Trudy Matematicheskogo Instituta Imeni V. A. Steklova 292 (2016) 264–267.
S. V. Tikhonov, On genus of division algebra, Manuscripta Mathematica, https://doi.org/10.1007/s00229-020-01184-4.
R. Treger, Reflexive modules, Journal of Algebra 54 (1978) 444–446.
A. R. Wadsworth, Valuation theory on finite dimensional division algebras, in Valuation Theory and its Applications. Vol. I (Saskatoon, SK, 1999), Fields Institute Communications, Vol. 32, American Mathematical Society, Providence, RI, 2002, pp. 385–449.
C. Weibel, The K-Book, Graduate Studies in Mathematics, Vol. 145, American Mathematical Society, Providence, RI, 2013.
A. Yamasaki, Strong approximation theorem for division algebras over R(X), Journal of the Mathematical Society of Japan 49 (1997) 455–467.
Acknowledgements
The first author was supported by an NSERC research grant. During the preparation of the paper, the second author visited Princeton University and the Institute for Advanced Study on a Simons Fellowship; the hospitality of both institutions and the generous support of the Simons Foundation are thankfully acknowledged. The third author was partially supported by an AMS-Simons Travel Grant. We would like to thank the anonymous referee for offering a number of corrections and valuable suggestions. We are also grateful to Louis Rowen for useful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
To Louis Rowen on the occasion of his retirement
Rights and permissions
About this article
Cite this article
Chernousov, V.I., Rapinchuk, A.S. & Rapinchuk, I.A. The finiteness of the genus of a finite-dimensional division algebra, and some generalizations. Isr. J. Math. 236, 747–799 (2020). https://doi.org/10.1007/s11856-020-1988-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-020-1988-x