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algebraic fundamental group

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Idea

The algebraic fundamental group is the fundamental group of a scheme, as defined by Grothendieck in SGA1. It is essentially the fundamental group as seen by étale homotopy, the étale fundamental group.

For fields this is essentially the Galois group.

In arithmetic geometry one also speaks of the arithmetic fundamental group.

Definition

Let SS be a connected scheme. Recall that a finite étale cover of SS is a finite flat surjection XSX\to S such that each fibre at a point sSs \in S is the spectrum of a finite étale algebra over the local ring at ss. Fix a geometric point s¯:Spec(Ω)Ω\overline{s} : Spec(\Omega) \to \Omega.

For a finite étale cover, XSX\to S, we consider the geometric fibre, X× SSpec(Ω)X\times_S Spec (\Omega), over s¯\overline{s}, and denote by Fib s¯(X)Fib_\overline{s} (X) its underlying set. This gives a set-valued functor on the category of finite étale covers of XX.

The algebraic fundamental group, π 1(S,s¯)\pi_1(S, \overline{s}) is defined to be the automorphism group of this functor.

For more on this area, see at étale homotopy.

(This entry is a stub and needs more work, including the linked entries that do not yet exist! Also explanation of Ω\Omega. It is adapted from the first reference below.)

References

or in a lengthier form:

  • Tamás Szamuely, Galois Groups and Fundamental Groups, Cambridge Studies in Advanced Mathematics, vol. 117, Cambridge University Press, 2009.

An earlier version is to be found here.

A paper on a closely related subject is

See also

Revised on September 1, 2014 11:09:26 by Urs Schreiber (185.37.147.14)