## A note on the cohomology of the Langlands group

HTML articles powered by AMS MathViewer

- by Edward S.T. Fan; with an appendix by M. Flach PDF
- Trans. Amer. Math. Soc.
**367**(2015), 2905-2920 Request permission

## Abstract:

We begin with a comparison of various cohomology theories for topological groups. Using the continuity result for Moore cohomology, we establish a Hochschild-Serre spectral sequence for a slightly larger class of groups. We use these properties to compute the cohomology of the Langlands group of a totally imaginary field. The appendix answers a question raised by Flach concerning the cohomological dimension of the group $\mathbb {R}$.## References

- James Arthur,
*A note on the automorphic Langlands group*, Canad. Math. Bull.**45**(2002), no. 4, 466–482. Dedicated to Robert V. Moody. MR**1941222**, DOI 10.4153/CMB-2002-049-1 - T. Austin,
*Continuity properties of Moore cohomology*, Preprint, available online at, arXiv.org:1004.4937, 2010. - Kenneth S. Brown,
*Cohomology of groups*, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR**672956**, DOI 10.1007/978-1-4684-9327-6 - N. Bourbaki,
*General Topology*, Springer-Verlag, Berlin (1989). - M. Flach,
*Cohomology of topological groups with applications to the Weil group*, Compos. Math.**144**(2008), no. 3, 633–656. MR**2422342**, DOI 10.1112/S0010437X07003338 - R. Gonzales,
*Localization in equivariant cohomology and GKM theory*, http://www.math. uwo.ca/ rgonzal3/qfy.pdf. - A. Grothendieck, M. Artin, J. L. Verdier,
*Theorie des topos et cohomologie etale des schemas (SGA4)*, Lecture Notes in Math. Soc.**269, 270, 271**, Springer, Berlin (1972). - Karl H. Hofmann and Sidney A. Morris,
*The Lie theory of connected pro-Lie groups*, EMS Tracts in Mathematics, vol. 2, European Mathematical Society (EMS), Zürich, 2007. A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups. MR**2337107**, DOI 10.4171/032 - Karl H. Hofmann and Paul S. Mostert,
*Cohomology theories for compact abelian groups*, Springer-Verlag, New York-Heidelberg; VEB Deutscher Verlag der Wissenschaften, Berlin, 1973. With an appendix by Eric C. Nummela. MR**0372113**, DOI 10.1007/978-3-642-80670-4 - Stephen Lichtenbaum,
*The Weil-étale topology for number rings*, Ann. of Math. (2)**170**(2009), no. 2, 657–683. MR**2552104**, DOI 10.4007/annals.2009.170.657 - Saunders Mac Lane and Ieke Moerdijk,
*Sheaves in geometry and logic*, Universitext, Springer-Verlag, New York, 1994. A first introduction to topos theory; Corrected reprint of the 1992 edition. MR**1300636**, DOI 10.1007/978-1-4612-0927-0 - Calvin C. Moore,
*Extensions and low dimensional cohomology theory of locally compact groups. I, II*, Trans. Amer. Math. Soc.**113**(1964), 40–63; ibid. 113 (1964), 64–86. MR**171880**, DOI 10.1090/S0002-9947-1964-0171880-5 - Calvin C. Moore,
*Group extensions and cohomology for locally compact groups. III*, Trans. Amer. Math. Soc.**221**(1976), no. 1, 1–33. MR**414775**, DOI 10.1090/S0002-9947-1976-0414775-X - Paul S. Mostert,
*Local cross sections in locally compact groups*, Proc. Amer. Math. Soc.**4**(1953), 645–649. MR**56614**, DOI 10.1090/S0002-9939-1953-0056614-5 - Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg,
*Cohomology of number fields*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2000. MR**1737196** - John F. Price,
*Lie groups and compact groups*, London Mathematical Society Lecture Note Series, No. 25, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR**0450449**, DOI 10.1017/CBO9780511600715 - Markus Stroppel,
*Locally compact groups*, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2006. MR**2226087**, DOI 10.4171/016 - David Wigner,
*Algebraic cohomology of topological groups*, Trans. Amer. Math. Soc.**178**(1973), 83–93. MR**338132**, DOI 10.1090/S0002-9947-1973-0338132-7

## Additional Information

**Edward S.T. Fan**- Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
- Email: sfan@caltech.edu
**M. Flach**- Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
- Received by editor(s): September 3, 2012
- Received by editor(s) in revised form: June 12, 2013
- Published electronically: February 25, 2014
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**367**(2015), 2905-2920 - MSC (2010): Primary 11F75, 14F20; Secondary 20J06, 22A99
- DOI: https://doi.org/10.1090/S0002-9947-2014-06230-2
- MathSciNet review: 3301886