Serre's conjecture II (algebra)
- 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.
In mathematics, Jean-Pierre Serre conjectured 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.
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.) The conjecture also holds when F is finitely generated over the complex numbers and has transcendence degree at most 2.
The conjecture is also known to hold for certain groups G. For special linear groups, it is a consequence of the Merkurjev–Suslin theorem. Building on this result, the conjecture holds if G is a classical group. The conjecture also holds if G is one of certain kinds of exceptional group.
- Serre, J-P. (1962). "Cohomologie galoisienne des groupes algébriques linéaires". Colloque sur la théorie des groupes algébriques: 53–68.
- Serre, J-P. (1964). Cohomologie galoisienne. Lecture Notes in Mathematics 5. Springer.
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- Merkurjev, A.S.; Suslin, A.A. (1983). "K-cohomology of Severi-Brauer varieties and the norm-residue homomorphism". Math. USSR Izvestiya 21: 307–340. doi:10.1070/im1983v021n02abeh001793.
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- Gille, P. (2001). "Cohomologie galoisienne des groupes algebriques quasi-déployés sur des corps de dimension cohomologique ≤ 2". Compositio Mathematica 125: 283–325.