mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Jean-Pierre Serre conjectured the following statement regarding the Galois cohomology of a

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...

semisimple algebraic group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...

. Namely, he conjectured that if ''G'' is such a group over a perfect field ''F'' of

cohomological dimension In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory. Cohomological ...

at most 2, then the Galois cohomology set ''H''¹(''F'', ''G'') is zero. A converse of the conjecture holds: if the field ''F'' is perfect and if the cohomology set ''H''¹(''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 In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ...

) or a

global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...

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 In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor ''K''-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of ...

. 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 In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...

.{{cite journal, last=Gille, first=P., title=Cohomologie galoisienne des groupes algebriques quasi-déployés sur des corps de dimension cohomologique ≤ 2, journal=Compositio Mathematica, year=2001, volume=125, issue=3, pages=283–325, doi=10.1023/A:1002473132282, s2cid=124765999, doi-access=free

References

External links

Philippe Gille's survey of the conjecture
Field (mathematics) Algebraic number theory Unsolved problems in number theory