Fundamental group
Étale fundamental group

In algebraic geometry, the so-called étale fundamental group is used as a replacement for the fundamental group. Since the Zariski topology on an algebraic variety or scheme X is much coarser than, say, the topology of open subsets in R n, {\displaystyle \mathbb {R} ^{n},} it is no longer meaningful to consider continuous maps from an interval to X. Instead, the approach developed by Grothendieck consists in constructing π 1 et {\displaystyle \pi _{1}^{\text{et}}} by considering all finite étale covers of X. These serve as an algebro-geometric analogue of coverings with finite fibers.

This yields a theory applicable in situations where no great generality classical topological intuition whatsoever is available, for example for varieties defined over a finite field. Also, the étale fundamental group of a field is its (absolute) Galois group. On the other hand, for smooth varieties X over the complex numbers, the étale fundamental group retains much of the information inherent in the classical fundamental group: the former is the profinite completion of the latter.
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