Grothendieck connection explained

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

M be a manifold and \pi : E \to M a surjective submersion, so that E is a manifold fibred over M. Let J ^{1(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 ) J ^{1(M,} E) \to 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

\Delta : M \to M x M. The sheaf I of ideals of \Delta in M x M consists of functions on M x M which vanish along the diagonal. Much of the infinitesimal geometry of M can be realized in terms of I. For instance, \Delta ^{*\left(I,} I ^{2\right)} is the sheaf of sections of the cotangent bundle. One may define a first-order infinitesimal neighborhood M ^{(2)} of \Delta in M x M to be the subscheme corresponding to the sheaf of ideals I ^{2.} (See below for a coordinate description.)

There are a pair of projections

p _{1,} p _{2} : M x M \to M given by projection the respective factors of the Cartesian product, which restrict to give projections p _{1,} p _{2} : M ^{(2)} \to M. One may now form the pullback of the fibre space E along one or the other of p _{1} or p _{2.} In general, there is no canonical way to identify p ^{*} _{1} E and p ^{*} _{2} E 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.

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.