en>Pwnzor.ak: Rerouted “4-dimensional” link in opening paragraph

2014-01-02T19:14:00Z

Rerouted “4-dimensional” link in opening paragraph

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			The approach of [[Alexander Grothendieck]] is concerned with the [[category theory\|category-theoretic]] properties that characterise the categories of finite ''G''-sets for a fixed [[profinite group]] ''G''. For example, ''G'' might be the group denoted <math>\hat{\Z}</math>, which is the [[inverse limit]] of the cyclic additive groups '''Z'''/n'''Z''' — or equivalently the completion of the [[infinite cyclic group]] '''Z''' for the topology of subgroups of finite [[Index of a subgroup\|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 classical [[Galois theory]] can be seen by regarding <math>\hat{\Z}</math> as the profinite Galois group Gal(<span style="text-decoration:overline">F</span>/F) of the [[algebraic closure]] <span style="text-decoration:overline">F</span> of any [[finite field]] ''F'', over ''F''. That is, the automorphisms of <span style="text-decoration:overline">F</span> fixing ''F'' are described by the inverse limit, as we take larger and larger finite [[splitting field]]s over ''F''. The connection with geometry can be seen when we look at [[covering space]]s of the [[unit disk]] in the [[complex plane]] with the origin removed: the finite covering realised by the ''z''<sup>''n''</sup> 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'' &cong; 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 in [[topos]] 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=André\|
			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=0-8218-2312-4
			}}

			* Borceux, F. and Janelidze, G., Cambridge University Press (2001). ''Galois theories'', ISBN 0-521-80309-8 (This book introduces the reader to the Galois theory of [[Grothendieck]], and some generalisations, leading to Galois [[groupoids]].)
			* Szamuely, T., Galois Groups and Fundamental Groups, Cambridge University Press, 2009.

			* Dubuc, E. J and de la Vega, C. S., On the Galois theory of Grothendieck, http://arxiv.org/abs/math/0009145v1

			[[Category:Field theory]]
			[[Category:Algebraic geometry]]
			[[Category:Category theory]]

en>Arthur Rubin: Reverted 3 edits by Puzzle314 (talk). (TW)

2012-04-07T05:51:22Z

Reverted 3 edits by Puzzle314 (talk). (TW)

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Magic tesseract - Revision history

en>Pwnzor.ak: Rerouted “4-dimensional” link in opening paragraph

en>Arthur Rubin: Reverted 3 edits by Puzzle314 (talk). (TW)