|
:''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 ''H''1(''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. ==References== 〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Serre's conjecture II (algebra)」の詳細全文を読む スポンサード リンク
|