Serre's conjecture II (algebra)
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 is zero.
A converse of the conjecture holds: if the field F is perfect and if the cohomology set H1 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 or a global field with no real embeddings ). This is a special case of the Kneser–Harder–Chernousov Hasse Principle for algebraic groups over global fields.
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.
