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algebraic fundamental group

Idea

This name is used for the fundamental group of a scheme, as defined by Grothendieck in SGA1.

Definition

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

For a finite étale cover, $X\to S$, we consider the geometric fibre, $X{\times }_S\mathrm{Spec}(\Omega )$, over $\overline{s}$, and denote by ${\mathrm{Fib}}_{\overline{s}}(X)$ its underlying set. This gives a set-valued functor on the category of finite étale covers of $X$.

The algebraic fundamental group, ${\pi }_1(S,\overline{s})$ 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 $\Omega$. It is adapted from the first reference below.)

References

• Tamás Szamuely, Heidelberg Lectures on Fundamental Groups, 2010.

or in a lengthier form:

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

and earlier version is to be found here.