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.

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

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

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

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.