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.
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.
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(√)). 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.
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- Serre, Jean-Pierre (1995). "Cohomologie galoisienne : progrès et problèmes". Astérisque. 227: 229–247. MR 1321649. Zbl 0837.12003 – via NUMDAM.
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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.