title: Grothendieck connection
text: # 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 $M$ be a manifold and $\pi :E\to M$ a surjective submersion, so that $E$ is a manifold fibred over $M.$ Let $J^{1}(M,E)$ be the first-order jet bundle of sections of $E.$ This may be regarded as a bundle over $M$ or a bundle over the total space of $E.$ With the latter interpretation, an Ehresmann connection is a section of the bundle (over $E$) $J^{1}(M,E)\to E.$ 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 $\Delta :M\to M\times M.$ The sheaf $I$ of ideals of $\Delta$ in $M\times M$ consists of functions on $M\times M$ which vanish along the diagonal. Much of the infinitesimal geometry of $M$ can be realized in terms of $I.$ For instance, $\Delta ^{*}\left(I,I^{2}\right)$ is the sheaf of sections of the cotangent bundle. One may define a first-order infinitesimal neighborhood $M^{(2)}$ of $\Delta$ in $M\times M$ to be the subscheme corresponding to the sheaf of ideals $I^{2}.$ (See below for a coordinate description.)
  
  There are a pair of projections $p_{1},p_{2}:M\times M\to M$ given by projection the respective factors of the Cartesian product, which restrict to give projections $p_{1},p_{2}:M^{(2)}\to M.$ One may now form the pullback of the fibre space $E$ along one or the other of $p_{1}$ or $p_{2}.$ In general, there is no canonical way to identify $p_{1}^{*}E$ and $p_{2}^{*}E$ 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.
  
  