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.

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 (2008). "Families of rationally simply connected varieties over surfaces and torsors for semisimple groups". arXiv:0809.5224 [math.AG].
  5. Merkurjev, A.S.; Suslin, A.A. (1983). "K-cohomology of Severi-Brauer varieties and the norm-residue homomorphism". Math. USSR Izvestiya. 21 (2): 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. S2CID 124673233.
  7. Gille, P. (2001). "Cohomologie galoisienne des groupes algebriques quasi-déployés sur des corps de dimension cohomologique ≤ 2". Compositio Mathematica. 125 (3): 283–325. doi:10.1023/A:1002473132282. S2CID 124765999.
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