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.

be a manifold a surjective submersion be the first-order jet bundle The problem is thus to obtain an intrinsic description of the sheaf of sections of this vector bundle.

is the sheaf of sections of the cotangent bundle. One may define a first-order infinitesimal neighborhood to be the subscheme (See below for a coordinate description.)

One may now form the pullback 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

References

• Osserman, B., "Connections, curvature, and p-curvature", preprint.

• Katz, N., "Nilpotent connections and the monodromy theorem", IHES Publ. Math. 39 (1970) 175–232.