Étale fundamental group

Étale fundamental group

The étale fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental 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 modulo homotopy. 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 an algebraic curve is not appropriate either, since in the most familiar case (over the complex numbers) such a "loop" has two real dimensions 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 of deck transformations of the universal covering space. This is more promising: finite étale morphisms 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 an inverse limit of automorphism 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 is pro-representable, in fact by "Galois covers" of X; this means that we have a projective system {X_j o X_i mid i < j in I} indexed by a directed 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 a compact Riemann surface, or more general complex non-singular complete variety "V". The algebraic fundamental group, as it is typically called in this case, is the profinite completion of &pi;1("V").

Notes

ee also

* étale morphism
* topological space
* fundamental group
* classification of covering spaces

----


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Fundamental group — In mathematics, the fundamental group is one of the basic concepts of algebraic topology. Associated with every point of a topological space there is a fundamental group that conveys information about the 1 dimensional structure of the portion of …   Wikipedia

  • Group (mathematics) — This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines …   Wikipedia

  • Étale cohomology — In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil… …   Wikipedia

  • Étale morphism — In algebraic geometry, a field of mathematics, an étale morphism (pronunciation IPA|) is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem,… …   Wikipedia

  • Étale Fundamentalgruppe — Die Étale Fundamentalgruppe wird in der algebraischen Geometrie untersucht. Sie ist ein Analogon der Fundamentalgruppe topologischer Räume für Schemata. Sie verallgemeinert den Begriff der Galoisgruppe und wurde von Alexander Grothendieck und… …   Deutsch Wikipedia

  • Group cohomology — This article is about homology and cohomology of a group. For homology or cohomology groups of a space or other object, see Homology (mathematics). In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well… …   Wikipedia

  • List of mathematics articles (E) — NOTOC E E₇ E (mathematical constant) E function E₈ lattice E₈ manifold E∞ operad E7½ E8 investigation tool Earley parser Early stopping Earnshaw s theorem Earth mover s distance East Journal on Approximations Eastern Arabic numerals Easton s… …   Wikipedia

  • Groupe (mathématiques) — Pour les articles homonymes, voir Groupe. Les manipulations possibles du cube de Rubik forment un groupe. En mathématiques, un groupe est un ensemble …   Wikipédia en Français

  • Langlands program — The Langlands program is a web of far reaching and influential conjectures that connect number theory and the representation theory of certain groups. It was proposed by Robert Langlands beginning in 1967. Connection with number theory The… …   Wikipedia

  • Enriques–Kodaira classification — In mathematics, the Enriques–Kodaira classification is a classification of compact complex surfaces into ten classes. For each of these classes, the surfaces in the class can be parametrized by a moduli space. For most of the classes the moduli… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”