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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 π : ''E'' → ''M'' a surjective submersion, so that ''E'' is a manifold fibred over ''M''. Let J1(''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'') J1(''M'',''E'') → ''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''. The sheaf ''I'' of ideals of Δ in ''M'' × ''M'' consists of functions on ''M'' × ''M'' which vanish along the diagonal. Much of the infinitesimal geometry of ''M'' can be realized in terms of ''I''. For instance, Δ * (''I''/''I''2) is the sheaf of sections of the cotangent bundle. One may define a ''first-order infinitesimal neighborhood'' ''M''(2) of Δ in ''M'' × ''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'' × ''M'' → ''M'' given by projection the respective factors of the Cartesian product, which restrict to give projections ''p''1, ''p''2 : ''M''(2) → ''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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Grothendieck connection」の詳細全文を読む スポンサード リンク
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