Serre's conjecture II (algebra)

In mathematics, Jean-Pierre Serre conjectured[1][2] the following statement regarding the Galois cohomology of a simply connected semisimple algebraic group. Namely, he conjectured that if G is such a group over a perfect field F of cohomological dimension at most 2, then the Galois cohomology set H1(F, G) is zero.

Not to be confused with the Serre conjecture in number theory or the Quillen–Suslin theorem, which is sometimes also referred to as Serre's conjecture.

A converse of the conjecture holds: if the field F is perfect and if the cohomology set H1(F, G) is zero for every semisimple simply connected algebraic group G then the p-cohomological dimension of F is at most 2 for every prime p.[3]

The conjecture holds in the case where F is a local field (such as p-adic field) or a global field with no real embeddings (such as Q(√−1)). This is a special case of the Kneser–Harder–Chernousov Hasse Principle for algebraic groups over global fields. (Note that such fields do indeed have cohomological dimension at most 2.[2]) The conjecture also holds when F is finitely generated over the complex numbers and has transcendence degree at most 2.[4]

The conjecture is also known to hold for certain groups G. For special linear groups, it is a consequence of the Merkurjev–Suslin theorem.[5] Building on this result, the conjecture holds if G is a classical group.[6] The conjecture also holds if G is one of certain kinds of exceptional group.[7]

References
  1. Serre, J-P. (1962). "Cohomologie galoisienne des groupes algébriques linéaires". Colloque sur la théorie des groupes algébriques: 53–68.
  2. Serre, J-P. (1964). Cohomologie galoisienne. Lecture Notes in Mathematics. 5. Springer.
  3. Serre, Jean-Pierre (1995). "Cohomologie galoisienne : progrès et problèmes". Astérisque. 227: 229–247. MR 1321649. Zbl 0837.12003 – via NUMDAM.
  4. de Jong, A.J.; He, Xuhua; Starr, Jason Michael. "Families of rationally simply connected varieties over surfaces and torsors for semisimple groups". arXiv:0809.5224.
  5. Merkurjev, A.S.; Suslin, A.A. (1983). "K-cohomology of Severi-Brauer varieties and the norm-residue homomorphism". Math. USSR Izvestiya. 21: 307–340. Bibcode:1983IzMat..21..307M. doi:10.1070/im1983v021n02abeh001793.
  6. Bayer-Fluckiger, E.; Parimala, R. (1995). "Galois cohomology of the classical groups over fields of cohomological dimension ≤ 2". Inventiones Mathematicae. 122: 195–229. Bibcode:1995InMat.122..195B. doi:10.1007/BF01231443.
  7. Gille, P. (2001). "Cohomologie galoisienne des groupes algebriques quasi-déployés sur des corps de dimension cohomologique ≤ 2". Compositio Mathematica. 125: 283–325.
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