- Étale fundamental group
The étale fundamental group is an analogue in
algebraic geometry , for schemes, of the usualfundamental group of topological spaces.Topological analogue
In
algebraic topology , the fundamental group:pi_1(T)
of a
connected topological space T is defined to be the group of loops based at a point modulohomotopy . When one wants to obtain something similar in the algebraic category, this definition encounters problems.One cannot simply attempt to use the same definition, since the result will be wrong if one is working in positive characteristic. More to the point, the topology on a
scheme fails to capture much of the structure of the scheme. Simply choosing the "loop" to be analgebraic curve is not appropriate either, since in the most familiar case (over thecomplex number s) such a "loop" has two realdimension s rather than one.Covering spaces
This discussion follows Milne [ [http://www.jmilne.org/math/CourseNotes/math732.html James Milne, "Lectures on Étale Cohomology"] (online course notes)] .
In the
classification of covering spaces , it is shown that the fundamental group is exactly the group ofdeck transformation s of theuniversal covering space . This is more promising: finiteétale morphism s are the appropriate generalization of covering spaces. Unfortunately, the universal covering space is often an "infinite" covering of the original space, which is unlikely to yield anything manageable in the algebraic category. Finite coverings, on the other hand are tractable, so one can define the algebraic fundamental group as aninverse limit ofautomorphism groups.Let X be a scheme, let x be a
geometric point of X, and let C be the category of pairs Y,f) such that f colon Y o X is a finite étale morphism ("finite étale schemes over X"). Morphisms Y,f) o (Y',f') in this category are morphisms Y o Y' as schemes over X. This category has a natural functor given x, namely the functor:F(Y) = operatorname{Hom}_X(x, Y);
geometrically this is the fiber of Y o X over x, and abstractly it is the covariant
Yoneda functor "co-represented" by x. The quotation marks are because x o X is not in fact a finite étale morphism, so that F is not actually representable (in general). However, it ispro-representable , in fact by "Galois covers" of X; this means that we have aprojective system X_j o X_i mid i < j in I} indexed by adirected set I, where the X_i are of course finite étale schemes over X,:operatorname{Aut}_X(X_i) = operatorname{deg}(X_i/X), and:F(Y) = varinjlim_{i in I} operatorname{Hom}_C(X_i, Y):(the subscript C is to emphasize that this Hom-set is in the category C).
Note that for two such X_i, X_j the map X_j o X_i induces a group homomorphism
:operatorname{Aut}_X(X_j) o operatorname{Aut}_X(X_i)
which produces a projective system of automorphism groups from the projective system X_i}. We then make the following definition: the "étale fundamental group" pi_1(X,x) of X at x is the inverse limit
:pi_1(X,x) = varprojlim_{i in I} {operatorname{Aut_X(X_i).
GAGA results
The general comparison machinery called
GAGA gives the connection in the case of acompact Riemann surface , or more general complexnon-singular complete variety "V". The algebraic fundamental group, as it is typically called in this case, is theprofinite completion of π1("V").Notes
ee also
*
étale morphism
*topological space
*fundamental group
*classification of covering spaces ----
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