Grothendieck's Galois theory

Grothendieck's Galois theory

In mathematics, Grothendieck's Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. It provides, in the classical setting of field theory, an alternative perspective to that of Emil Artin based on linear algebra, which became standard from about the 1930s.

The approach of Alexander Grothendieck is concerned with the category theory properties that characterise the categories of "finite G-sets" for G a fixed profinite group. For example G might be the group denoted hat{}, which is the inverse limit of the cyclic additive groups Z/nZ - or equivalently the completion of the infinite cyclic group Z for the topology of subgroups of finite index. A finite G-set is then a finite set X on which G acts through a quotient finite cyclic group; so that it is specified by giving some permutation of X.

In the above example, a connection with the classical Galois theory can be seen by regarding hat{} as the pro-finite Galois group Gal(F/F) of the algebraic closure F of any finite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F. The connection with geometry can be seen when we look at covering spaces of the unit disk in the complex plane with the origin removed: the finite covering realised by the "z""n" map of the disk, thought of by means of a complex number variable "z", corresponds to the subgroup "n".Z of the fundamental group of the punctured disk.

The theory of Grothendieck, published in SGA1, shows how to reconstruct the category of G-sets from a "fibre functor" Φ, which in the geometric setting takes the fibre of a covering above a fixed base point (as a set). In fact there is an isomorphism proved of the type

:"G" ≅ "Aut"(Φ),

the latter being the group of automorphisms (self-natural equivalences) of Φ. An abstract classification of categories with a functor to the category of sets is given, by means of which one can recognise categories of G-sets for G profinite.

To see how this applies to the case of fields, one has to study the tensor product of fields. Later developments in topos theory make this all part of a theory of "atomic toposes".

References

* cite book
last=Grothendieck
first=A.
coauthors=et al.
title=SGA1 "Revêtements étales et groupe fondamental, 1960–1961'
series=Lecture Notes in Mathematics 224
year=1971
publisher=Springer Verlag
* cite book
last=Joyal
first=Andre
coauthors= Tierney, Myles
title=An Extension of the Galois Theory of Grothendieck
series=Memoirs of the American Mathematical Society
year= 1984
publisher=Proquest Info & Learning
isbn=0821823124

* Borceux, F. and Janelidze, G., Cambridge University Press (2001). "Galois theories", ISBN 0521803098 (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.)


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