Grothendieck connection

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.

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 connection 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 [math]\displaystyle{ M }[/math] be a manifold and [math]\displaystyle{ \pi : E \to M }[/math] a surjective submersion, so that [math]\displaystyle{ E }[/math] is a manifold fibred over [math]\displaystyle{ M. }[/math] Let [math]\displaystyle{ J^1(M, E) }[/math] be the first-order jet bundle of sections of [math]\displaystyle{ E. }[/math] This may be regarded as a bundle over [math]\displaystyle{ M }[/math] or a bundle over the total space of [math]\displaystyle{ E. }[/math] With the latter interpretation, an Ehresmann connection is a section of the bundle (over [math]\displaystyle{ E }[/math]) [math]\displaystyle{ J^1(M, E) \to E. }[/math] 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 [math]\displaystyle{ \Delta : M \to M \times M. }[/math] The sheaf [math]\displaystyle{ I }[/math] of ideals of [math]\displaystyle{ \Delta }[/math] in [math]\displaystyle{ M \times M }[/math] consists of functions on [math]\displaystyle{ M \times M }[/math] which vanish along the diagonal. Much of the infinitesimal geometry of [math]\displaystyle{ M }[/math] can be realized in terms of [math]\displaystyle{ I. }[/math] For instance, [math]\displaystyle{ \Delta^*\left(I, I^2\right) }[/math] is the sheaf of sections of the cotangent bundle. One may define a first-order infinitesimal neighborhood [math]\displaystyle{ M^{(2)} }[/math] of [math]\displaystyle{ \Delta }[/math] in [math]\displaystyle{ M \times M }[/math] to be the subscheme corresponding to the sheaf of ideals [math]\displaystyle{ I^2. }[/math] (See below for a coordinate description.)

There are a pair of projections [math]\displaystyle{ p_1, p_2 : M \times M \to M }[/math] given by projection the respective factors of the Cartesian product, which restrict to give projections [math]\displaystyle{ p_1, p_2 : M^{(2)} \to M. }[/math] One may now form the pullback of the fibre space [math]\displaystyle{ E }[/math] along one or the other of [math]\displaystyle{ p_1 }[/math] or [math]\displaystyle{ p_2. }[/math] In general, there is no canonical way to identify [math]\displaystyle{ p_1^* E }[/math] and [math]\displaystyle{ p_2^* E }[/math] with each other. A Grothendieck connection is a specified isomorphism between these two spaces. One may proceed to define curvature and p-curvature of a connection in the same language.

See also

• Connection (mathematics) – Function which tells how a certain variable changes as it moves along certain points in space

References

1. Osserman, B., "Connections, curvature, and p-curvature", preprint.
2. Katz, N., "Nilpotent connections and the monodromy theorem", IHES Publ. Math. 39 (1970) 175–232.