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 In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space ''S'' of a family of algebraic varieties V_s. The fibers of the vector bundle are the de Rham cohomology groups H^k_(V_s) of the fibers V_s o ...

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 In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the ...

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 Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper se ...

on a

Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is cal ...

. 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 In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. ...

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

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^

\Delta

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^ \to M.

One may now form the

pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...

of the fibre space

E

along one or the other of

p_1

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.

References

# Osserman, B., "Connections, curvature, and p-curvature", ''preprint''. # Katz, N., "Nilpotent connections and the monodromy theorem", ''IHES Publ. Math.'' 39 (1970) 175–232. {{Manifolds Connection (mathematics) Algebraic geometry

Introduction and motivation

See also

References