Grothendieck connection
Encyclopedia
In algebraic geometry
and synthetic differential geometry
, a Grothendieck connection is a way of viewing connections
in terms of descent data from infinitesimal neighbourhoods of the diagonal.
generalizes the Koszul connection. The construction itself must satisfy a requirement of geometric invariance, which may be regarded as the analog of covariance for a wider class of structures including the schemes
of algebraic geometry. Thus the connection in a certain sense must live in a natural sheaf
on a Grothendieck topology
. In this section, we discuss how to describe an Ehresmann connection in sheaf-theoretic terms as a Grothendieck connection.
Let M be a manifold and π : E → M a surjective submersion
, so that E is a manifold fibred over M. Let J1(M,E) be the first-order jet bundle
of sections of E. This may be regarded as a bundle over M or a bundle over the total space of E. With the latter interpretation, an Ehresmann connection is a section of the bundle (over E) J1(M,E) → E. The problem is thus to obtain an intrinsic description of the sheaf of sections of this vector bundle.
Grothendieck's solution is to consider the diagonal embedding Δ : M → M × M. The sheaf I of ideals of Δ in M × M consists of functions on M × M which vanish along the diagonal. Much of the infinitesimal geometry of M can be realized in terms of I. For instance, Δ* (I/I2) is the sheaf of sections of the cotangent bundle
. One may define a first-order infinitesimal neighborhood M(2) of Δ in M × M to be the subscheme
corresponding to the sheaf of ideals I2. (See below for a coordinate description.)
There are a pair of projections p1, p2 : M × M → M given by projection the respective factors of the Cartesian product, which restrict to give projections p1, p2 : M(2) → M. One may now form the pullback
of the fibre space E along one or the other of p1 or p2. In general, there is no canonical way to identify p1*E and p2*E with each other. A Grothendieck connection is a specified isomorphism between these two spaces.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and synthetic differential geometry
Synthetic differential geometry
In mathematics, synthetic differential geometry is a reformulation of differential geometry in the language of topos theory, in the context of an intuitionistic logic characterized by a rejection of the law of excluded middle. There are several insights that allow for such a reformulation...
, a Grothendieck connection is a way of viewing connections
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
in terms of descent data from infinitesimal neighbourhoods of the diagonal.
Introduction and motivation
The Grothendieck connection is a generalization of the Gauss-Manin connection constructed in a manner analogous to that in which the Ehresmann connectionEhresmann connection
In differential geometry, an Ehresmann connection is a version of the notion of a connection, which makes sense on any smooth fibre bundle...
generalizes the Koszul connection. The construction itself must satisfy a requirement of geometric invariance, which may be regarded as the analog of covariance for a wider class of structures including the schemes
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
of algebraic geometry. Thus the connection in a certain sense must live in a natural sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
on a Grothendieck topology
Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
. In this section, we discuss how to describe an Ehresmann connection in sheaf-theoretic terms as a Grothendieck connection.
Let M be a manifold and π : E → M a surjective submersion
Submersion (mathematics)
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology...
, so that E is a manifold fibred over M. Let J1(M,E) be the first-order jet bundle
Jet bundle
In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form...
of sections of E. This may be regarded as a bundle over M or a bundle over the total space of E. With the latter interpretation, an Ehresmann connection is a section of the bundle (over E) J1(M,E) → E. The problem is thus to obtain an intrinsic description of the sheaf of sections of this vector bundle.
Grothendieck's solution is to consider the diagonal embedding Δ : M → M × M. The sheaf I of ideals of Δ in M × M consists of functions on M × M which vanish along the diagonal. Much of the infinitesimal geometry of M can be realized in terms of I. For instance, Δ* (I/I2) is the sheaf of sections of the cotangent bundle
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
. One may define a first-order infinitesimal neighborhood M(2) of Δ in M × M to be the subscheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
corresponding to the sheaf of ideals I2. (See below for a coordinate description.)
There are a pair of projections p1, p2 : M × M → M given by projection the respective factors of the Cartesian product, which restrict to give projections p1, p2 : M(2) → M. One may now form the pullback
Pullback bundle
In mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...
of the fibre space E along one or the other of p1 or p2. In general, there is no canonical way to identify p1*E and p2*E with each other. A Grothendieck connection is a specified isomorphism between these two spaces.