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 {\displaystyle M} be a manifold and π : E → M {\displaystyle \pi :E\to M} a surjective submersion, so that E {\displaystyle E} is a manifold fibred over M . {\displaystyle M.} Let J 1 ( M , E ) {\displaystyle J^{1}(M,E)} be the first-order jet bundle of sections of E . {\displaystyle E.} This may be regarded as a bundle over M {\displaystyle M} or a bundle over the total space of E . {\displaystyle E.} With the latter interpretation, an Ehresmann connection is a section of the bundle (over E {\displaystyle E} ) J 1 ( M , E ) → E . {\displaystyle 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 Δ : M → M × M . {\displaystyle \Delta :M\to M\times M.} The sheaf I {\displaystyle I} of ideals of Δ {\displaystyle \Delta } in M × M {\displaystyle M\times M} consists of functions on M × M {\displaystyle M\times M} which vanish along the diagonal. Much of the infinitesimal geometry of M {\displaystyle M} can be realized in terms of I . {\displaystyle I.} For instance, Δ ∗ ( I , I 2 ) {\displaystyle \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 ) {\displaystyle M^{(2)}} of Δ {\displaystyle \Delta } in M × M {\displaystyle M\times M} to be the subscheme corresponding to the sheaf of ideals I 2 . {\displaystyle I^{2}.} (See below for a coordinate description.) There are a pair of projections p 1 , p 2 : M × M → M {\displaystyle p_{1},p_{2}:M\times M\to M} given by projection the respective factors of the Cartesian product, which restrict to give projections p 1 , p 2 : M ( 2 ) → M . {\displaystyle p_{1},p_{2}:M^{(2)}\to M.} One may now form the pullback of the fibre space E {\displaystyle E} along one or the other of p 1 {\displaystyle p_{1}} or p 2 . {\displaystyle p_{2}.} In general, there is no canonical way to identify p 1 ∗ E {\displaystyle p_{1}^{*}E} and p 2 ∗ E {\displaystyle 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.
See also Connection (mathematics) – Function in mathematics
References
Osserman, B., "Connections, curvature, and p-curvature", preprint. Katz, N., "Nilpotent connections and the monodromy theorem", IHES Publ. Math. 39 (1970) 175–232.
