- Grothendieck's Galois theory
In
mathematics , Grothendieck's Galois theory is a highly abstract approach to theGalois theory of fields, developed around 1960 to provide a way to study thefundamental group ofalgebraic topology in the setting ofalgebraic geometry . It provides, in the classical setting of field theory, an alternative perspective to that ofEmil Artin based onlinear algebra , which became standard from about the 1930s.The approach of
Alexander Grothendieck is concerned with thecategory theory properties that characterise the categories of "finite G-sets" for G a fixedprofinite group . For example G might be the group denoted , which is theinverse limit of the cyclic additive groups Z/nZ - or equivalently the completion of theinfinite 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 as the pro-finite Galois group Gal(F/F) of thealgebraic closure F of anyfinite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finitesplitting field s over F. The connection with geometry can be seen when we look atcovering space s of theunit disk in thecomplex 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 equivalence s) 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 intopos theory make this all part of a theory of "atomic topos es".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 Galoisgroupoids .)
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