Serre's conjecture II

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.

Unsolved problem in mathematics

Must the Galois cohomology set of a simply connected semisimple algebraic group over a perfect field of cohomological dimension at most 2 always equal zero?

More unsolved problems in mathematics

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] .^[8]