In mathematics, the fundamental group scheme is a
group scheme
In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in ...
canonically attached to a
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
over a Dedekind scheme (e.g. the spectrum 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 ...
or the spectrum of a
discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R'' ...
). It is a generalisation of the
étale fundamental group The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
Topological analogue/informal discussion
In algebraic topology, the fundamental group ''π''1(''X' ...
. Although its existence was conjectured by
Alexander Grothendieck, the first proof if its existence is due, for schemes defined over fields, to
Madhav Nori.
A proof of its existence for schemes defined over Dedekind schemes is due to
Marco Antei
Marco may refer to:
People
* Marco (given name), people with the given name Marco
* Marco (actor) (born 1977), South Korean model and actor
* Georg Marco (1863–1923), Romanian chess player of German origin
* Tomás Marco (born 1942), Spanish co ...
, Michel Emsalem and Carlo Gasbarri.
History
The (topological)
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
associated with a
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 points ...
is 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 the
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es under
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
of the
loop
Loop or LOOP may refer to:
Brands and enterprises
* Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live
* Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets
* Loop Mobile, an ...
s contained in the space. Although it is still being studied for the classification of
algebraic varieties
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. Mo ...
even in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, for many applications the fundamental group has been found to be inadequate for the classification of objects, such as
schemes, that are more than just topological spaces. The same topological space may have indeed several distinct scheme structures, yet its topological fundamental group will always be the same. Therefore, it became necessary to create a new object that would take into account the existence of a structural
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
together with a topological space. This led to the creation of the
étale fundamental group The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
Topological analogue/informal discussion
In algebraic topology, the fundamental group ''π''1(''X' ...
, the projective limit of all finite groups acting on
étale covering In mathematics, more specifically in algebra, the adjective étale refers to several closely related concepts:
* Étale morphism
** Formally étale morphism
* Étale cohomology
* Étale topology
* Étale fundamental group
* Étale group scheme
...
s of the given scheme
. Nevertheless, in positive
characteristic the latter has obvious limitations, since it does not take into account the existence of
group scheme
In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in ...
s that are not étale (e.g.,
when the characteristic is
) and that act on
torsors over
, a natural generalization of the coverings. It was from this idea that Grothendieck hoped for the creation of a new true fundamental group (''un vrai groupe fondamental'', in French), the existence of which he conjectured, back in the early 1960s in his celebrated
SGA 1, Chapitre X. More than a decade had to pass before a first result on the existence of the fundamental group scheme came to light. As mentioned in the introduction this result was due to Madhav Nori who in 1976 published his first construction of this new object
for schemes defined over fields. As for the name he decided to abandon the ''true fundamental group'' name and he called it, as we know it nowadays, the ''fundamental group scheme''.
It is also often denoted as
, where
stands for Nori, in order to distinguish it from the previous fundamental groups and to its modern generalizations. The demonstration of the existence of
defined on regular schemes of dimension 1 had to wait about forty more years. There are various generalizations such as the ''
-fundamental group scheme''
and the ''quasi finite fundamental group scheme''
.
Definition and construction
The original definition and the first construction have been suggested by Nori for schemes
over fields. Then they have been adapted to a wider range of schemes. So far the only ''complete'' theories exist for schemes defined over schemes of dimension 0 (
spectra of fields) or dimension 1 (Dedekind schemes) so this is what will be discussed hereafter:
Definition
Let
be a Dedekind scheme (which can be the spectrum of a field) and
a faithfully
flat morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,
:f_P\colon \mathcal_ \t ...
, locally of finite type. Assume
has a section
. We say that
has a fundamental group scheme
if there exist a pro-finite and flat
-
torsor
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...
, with a section
such that for any finite
-torsor
with a section
there is a unique morphism of torsors
sending
to
.
Over a field
There are nowadays several existence results for the fundamental group scheme of a scheme
defined over a field
. Nori provides the first existence theorem when
is perfect and
is a proper morphism of schemes with
reduced and connected scheme. Assuming the existence of a section
, then the fundamental group scheme
of
in
is built as the affine group scheme naturally associated to the neutral
tannakian category
In mathematics, a Tannakian category is a particular kind of monoidal category ''C'', equipped with some extra structure relative to a given field ''K''. The role of such categories ''C'' is to approximate, in some sense, the category of linear re ...
(over
) of
essentially finite vector bundle In mathematics, an essentially finite vector bundle is a particular type of vector bundle defined by Madhav V. Nori, as the main tool in the construction of the fundamental group scheme. Even if the definition is not intuitive there is a nice char ...
s over
.
Nori also proves a that the fundamental group scheme exists when
is any field and
is any finite type, reduced and connected scheme over
. In this situation however there are no tannakian categories involved.
Since then several other existence results have been added, including some ''non reduced'' schemes.
Over a Dedekind scheme
Let
be a Dedekind scheme of dimension 1,
any connected scheme and
a faithfully flat morphism locally of finite type. Assume the existence of a section
. Then the existence of the fundamental group scheme
as a
group scheme
In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in ...
over
has been proved by
Marco Antei
Marco may refer to:
People
* Marco (given name), people with the given name Marco
* Marco (actor) (born 1977), South Korean model and actor
* Georg Marco (1863–1923), Romanian chess player of German origin
* Tomás Marco (born 1942), Spanish co ...
, Michel Emsalem and Carlo Gasbarri in the following situations:
* when for every
the fibres
are reduced
* when for every
the local ring
is integrally closed (e.g. when
is
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
).
Over a Dedekind scheme, however, there is no need to only consider
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 marked ...
group schemes: indeed
quasi-finite group schemes are also a very natural generalization of finite group schemes over fields. This is why Antei, Emsalem and Gasbarri also defined the quasi-finite fundamental group scheme
as follows:
let
be a Dedekind scheme and
a faithfully
flat morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,
:f_P\colon \mathcal_ \t ...
, locally of finite type. Assume
has a section
. We say that
has a quasi-finite fundamental group scheme
if there exist a pro-quasi-finite and flat
-
torsor
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...
, with a section
such that for any quasi-finite
-torsor
with a section
there is a unique morphism of torsors
sending
to
.
They proved the existence of
when for every
the fibres
are integral and normal.
Properties
Connections with the
étale fundamental group The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
Topological analogue/informal discussion
In algebraic topology, the fundamental group ''π''1(''X' ...
One can consider the largest pro-étale quotient of
. When the base scheme
is the spectrum of an algebraically closed field
then it ''coincides'' with the
étale fundamental group The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
Topological analogue/informal discussion
In algebraic topology, the fundamental group ''π''1(''X' ...
. More precisely the group of points
is isomorphic to
.
The product formula
For
and
any two smooth projective schemes over an algebraically closed field
the product formula holds, that is
.
[{{cite journal , doi=10.1007/s002220100191, title=On the fundamental group scheme , year=2002 , last1=Mehta , first1=V.B. , last2=Subramanian , first2=S. , journal=Inventiones Mathematicae , volume=148 , issue=1 , pages=143–150 , bibcode=2002InMat.148..143M , s2cid=121329868 ] This result was conjectured by Nori
and proved by
Vikram Mehta and Subramanian.
See also
*
Étale fundamental group
The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces.
Topological analogue/informal discussion
In algebraic topology, the fundamental group \pi_1(X,x) ...
*
Fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
Notes
Scheme theory
Topological methods of algebraic geometry