Serre's conjecture II

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($\sqrt{−1}$)). 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.