This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Grothendieck connection" – news · newspapers · books · scholar · JSTOR (May 2026) (Learn how and when to remove this message) This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (May 2026) (Learn how and when to remove this message) (Learn how and when to remove this message)

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 in mathematics

• Osserman, B., "Connections, curvature, and p-curvature", preprint.
• Katz, N., "Nilpotent connections and the monodromy theorem", IHES Publ. Math. 39 (1970) 175–232.