Let ${\displaystyle A}$  be a commutative ring and ${\displaystyle M}$  an A-module. There are different equivalent definitions of a connection on ${\displaystyle M}$ .^[2]

First definition edit

If ${\displaystyle k\to A}$  is a ring homomorphism, a ${\displaystyle k}$ -linear connection is a ${\displaystyle k}$ -linear morphism

${\displaystyle \nabla :M\to \Omega _{A/k}^{1}\otimes _{A}M}$ 

which satisfies the identity

${\displaystyle \nabla (am)=da\otimes m+a\nabla m}$ 

A connection extends, for all ${\displaystyle p\geq 0}$  to a unique map

${\displaystyle \nabla :\Omega _{A/k}^{p}\otimes _{A}M\to \Omega _{A/k}^{p+1}\otimes _{A}M}$ 

satisfying ${\displaystyle \nabla (\omega \otimes f)=d\omega \otimes f+(-1)^{p}\omega \wedge \nabla f}$ . A connection is said to be integrable if ${\displaystyle \nabla \circ \nabla =0}$ , or equivalently, if the curvature ${\displaystyle \nabla ^{2}:M\to \Omega _{A/k}^{2}\otimes M}$  vanishes.

Second definition edit

Let ${\displaystyle D(A)}$  be the module of derivations of a ring ${\displaystyle A}$ . A connection on an A-module ${\displaystyle M}$  is defined as an A-module morphism

${\displaystyle \nabla :D(A)\to \mathrm {Diff} _{1}(M,M);u\mapsto \nabla _{u}}$ 

such that the first order differential operators ${\displaystyle \nabla _{u}}$  on ${\displaystyle M}$  obey the Leibniz rule

${\displaystyle \nabla _{u}(ap)=u(a)p+a\nabla _{u}(p),\quad a\in A,\quad p\in M.}$ 

Connections on a module over a commutative ring always exist.

The curvature of the connection ${\displaystyle \nabla }$  is defined as the zero-order differential operator

${\displaystyle R(u,u')=[\nabla _{u},\nabla _{u'}]-\nabla _{[u,u']}\,}$ 

on the module ${\displaystyle M}$  for all ${\displaystyle u,u'\in D(A)}$ .

If ${\displaystyle E\to X}$  is a vector bundle, there is one-to-one correspondence between linear connections ${\displaystyle \Gamma }$  on ${\displaystyle E\to X}$  and the connections ${\displaystyle \nabla }$  on the ${\displaystyle C^{\infty }(X)}$ -module of sections of ${\displaystyle E\to X}$ . Strictly speaking, ${\displaystyle \nabla }$  corresponds to the covariant differential of a connection on ${\displaystyle E\to X}$ .

The notion of a connection on modules over commutative rings is straightforwardly extended to modules over a graded commutative algebra.^[3] This is the case of superconnections in supergeometry of graded manifolds and supervector bundles. Superconnections always exist.

If ${\displaystyle A}$  is a noncommutative ring, connections on left and right A-modules are defined similarly to those on modules over commutative rings.^[4] However these connections need not exist.

In contrast with connections on left and right modules, there is a problem how to define a connection on an R-S-bimodule over noncommutative rings R and S. There are different definitions of such a connection.^[5] Let us mention one of them. A connection on an R-S-bimodule ${\displaystyle P}$  is defined as a bimodule morphism

${\displaystyle \nabla :D(A)\ni u\to \nabla _{u}\in \mathrm {Diff} _{1}(P,P)}$ 

which obeys the Leibniz rule

${\displaystyle \nabla _{u}(apb)=u(a)pb+a\nabla _{u}(p)b+apu(b),\quad a\in R,\quad b\in S,\quad p\in P.}$ 

• Connection (vector bundle)
• Connection (mathematics)
• Noncommutative geometry
• Supergeometry
• Differential calculus over commutative algebras

• Sardanashvily, G. (2009). "Lectures on Differential Geometry of Modules and Rings". arXiv:0910.1515 [math-ph].