Connection (algebraic framework)
Geometry of quantum systems (e.g., noncommutative geometry and supergeometry) is mainly phrased in algebraic terms of modules and algebras. Connections on modules are generalization of a linear connection on a smooth vector bundle written as a Koszul connection on the -module of sections of . Commutative algebra Let be a commutative ring and an A-module. There are different equivalent definitions of a connection on . First definition If is a ring homomorphism, a -linear connection is a -linear morphism which satisfies the identity A connection extends, for all to a unique map satisfying . A connection is said to be integrable if , or equivalently, if the curvature vanishes. Second definition Let be the module of derivations of a ring . A connection on an A-module is defined as an A-module morphism such that the first order differential operators on obey the Leibniz rule Connections on a module over a commutative ring always exist. The curvature of the connection is defined as the zero-order differential operator on the module for all . If is a vector bundle, there is one-to-one correspondence between linear connections on and the connections on the -module of sections of . Strictly speaking, corresponds to the