 A
connection on an "A"-module formula_5 is defined
as an "A"-module morphism

such that the first order differential operators formula_11 on
formula_5 obey the Leibniz rule

Connections on a module over a commutative ring always exist.The curvature of the connection formula_14 is defined as
the zero-order differential operator

on the module formula_5 for all formula_17.If formula_18 is a vector bundle, there is one-to-one
correspondence between linear
connections formula_19 on formula_18 and the
connections formula_14 on the
formula_2-module of sections of formula_1.