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  be a manifold and  a surjective submersion, so that  is a manifold fibred over   Let  be the first-order jet bundle of sections of   This may be regarded as a bundle over  or a bundle over the total space of   With the latter interpretation, an Ehresmann connection is a section of the bundle (over )   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   The sheaf  of ideals of  in  consists of functions on  which vanish along the diagonal. Much of the infinitesimal geometry of  can be realized in terms of   For instance,  is the sheaf of sections of the cotangent bundle. One may define a first-order infinitesimal neighborhood  of  in  to be the subscheme corresponding to the sheaf of ideals   (See below for a coordinate description.) There are a pair of projections  given by projection the respective factors of the Cartesian product, which restrict to give projections   One may now form the pullback of the fibre space  along one or the other of  or   In general, there is no canonical way to identify  and  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. Connection (mathematics)
Algebraic geometry