étale fundamental group
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The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s.


Topological analogue/informal discussion

In
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, the fundamental group ''π''1(''X'',''x'') of a pointed topological space (''X'',''x'') is defined as the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of homotopy classes of loops based at ''x''. This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
with the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
. In the classification of covering spaces, it is shown that the fundamental group is exactly the group of
deck transformation A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete sp ...
s of the
universal covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
. This is more promising:
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
étale morphism In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy t ...
s are the appropriate analogue of covering spaces. Unfortunately, an algebraic variety ''X'' often fails to have a "universal cover" that is finite over ''X'', so one must consider the entire category of finite étale coverings of ''X''. One can then define the étale fundamental group as an
inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...
of finite automorphism groups.


Formal definition

Let X be a connected and locally
noetherian scheme In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. Thu ...
, let x be a
geometric point This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...
of X, and let C be the category of pairs (Y,f) such that f \colon Y \to X is a finite étale morphism from a scheme Y. Morphisms (Y,f)\to (Y',f') in this category are morphisms Y\to Y' as schemes over X. This category has a natural functor to the category of sets, namely the functor :F(Y) = \operatorname_X(x, Y); geometrically this is the fiber of Y \to X over x, and abstractly it is the
Yoneda functor In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewi ...
represented by x in the category of schemes over X. The functor F is typically not representable in C; however, it is pro-representable in C, in fact by Galois covers of X. This means that we have a projective system \ in C, indexed by a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements ha ...
I, where the X_i are Galois covers of X, i.e., finite étale schemes over X such that \#\operatorname_X(X_i) = \operatorname(X_i/X). It also means that we have given an isomorphism of functors :F(Y) = \varinjlim_ \operatorname_C(X_i, Y). In particular, we have a marked point P\in \varprojlim_ F(X_i) of the projective system. For two such X_i, X_j the map X_j \to X_i induces a group homomorphism \operatorname_X(X_j) \to \operatorname_X(X_i) which produces a projective system of automorphism groups from the projective system \. 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_ _X(X_i), with the inverse limit topology. The functor F is now a functor from C to the category of finite and continuous \pi_1(X,x)-sets, and establishes an ''equivalence of categories'' between C and the category of finite and continuous \pi_1(X,x)-sets.


Examples and theorems

The most basic example of a fundamental group is π1(Spec ''k''), the fundamental group of a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''k''. Essentially by definition, the fundamental group of ''k'' can be shown to be isomorphic to the absolute
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
Gal (''k''sep / ''k''). More precisely, the choice of a geometric point of Spec (''k'') is equivalent to giving a separably closed extension field ''K'', and the fundamental group with respect to that base point identifies with the Galois group Gal (''K'' / ''k''). This interpretation of the Galois group is known as
Grothendieck's Galois theory In mathematics, Grothendieck's Galois theory is an abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. It provides, in ...
. More generally, for any geometrically connected variety ''X'' over a field ''k'' (i.e., ''X'' is such that ''X''sep := ''X'' ×''k'' ''k''sep is connected) there is an
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
of profinite groups :1 → π1(''X''sep, ) → π1(''X'', ) → Gal(''k''sep / ''k'') → 1.


Schemes over a field of characteristic zero

For a scheme ''X'' that is of finite type over C, the complex numbers, there is a close relation between the étale fundamental group of ''X'' and the usual, topological, fundamental group of ''X''(C), the complex analytic space attached to ''X''. The algebraic fundamental group, as it is typically called in this case, is the profinite completion of π1(''X''). This is a consequence of the Riemann existence theorem, which says that all finite étale coverings of ''X''(C) stem from ones of ''X''. In particular, as the fundamental group of smooth curves over C (i.e., open Riemann surfaces) is well understood; this determines the algebraic fundamental group. More generally, the fundamental group of a proper scheme over any algebraically closed field of characteristic zero is known, because an extension of algebraically closed fields induces isomorphic fundamental groups.


Schemes over a field of positive characteristic and the tame fundamental group

For an algebraically closed field ''k'' of positive characteristic, the results are different, since Artin–Schreier coverings exist in this situation. For example, the fundamental group of the
affine line In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
\mathbf A^1_k is not topologically finitely generated. The ''tame fundamental group'' of some scheme ''U'' is a quotient of the usual fundamental group of ''U'' which takes into account only covers that are tamely ramified along ''D'', where ''X'' is some compactification and ''D'' is the complement of ''U'' in ''X''. For example, the tame fundamental group of the affine line is zero.


Affine schemes over a field of characteristic p

It turns out that every affine scheme X \subset \mathbf^n_k is a K(\pi,1)-space, in the sense that the etale homotopy type of X is entirely determined by its etale homotopy group. Note \pi = \pi_1^(X,\overline) where \overline is a geometric point.


Further topics

From a category-theoretic point of view, the fundamental group is a functor : → . The
inverse Galois problem In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved. There ...
asks what groups can arise as fundamental groups (or Galois groups of field extensions).
Anabelian geometry Anabelian geometry is a theory in number theory which describes the way in which the algebraic fundamental group ''G'' of a certain arithmetic variety ''X'', or some related geometric object, can help to restore ''X''. The first results for n ...
, for example Grothendieck's section conjecture, seeks to identify classes of varieties which are determined by their fundamental groups. studies higher étale homotopy groups by means of the étale homotopy type of a scheme.


The pro-étale fundamental group

have introduced a variant of the étale fundamental group called the ''pro-étale fundamental group''. It is constructed by considering, instead of finite étale covers, maps which are both étale and satisfy the valuative criterion of properness. For geometrically unibranch schemes (e.g., normal schemes), the two approaches agree, but in general the pro-étale fundamental group is a finer invariant: its profinite completion is the étale fundamental group.


See also

*
étale morphism In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy t ...
* Fundamental group *
Fundamental group scheme In mathematics, the fundamental group scheme is a group scheme canonically attached to a scheme over a Dedekind scheme (e.g. the spectrum of a field or the spectrum of a discrete valuation ring). It is a generalisation of the étale fundamental g ...


Notes


References

* * * * * {{DEFAULTSORT:Etale Fundamental Group Scheme theory Topological methods of algebraic geometry