# nLab connection for a differential graded algebra

Given a semi-free differential graded algebra ${\Omega }^{•}A$ over $k$-algebra $A$, a connection in $A$-module $M$ is a $k$-linear map

$\nabla :M{\otimes }_{A}{\Omega }^{•}A\to M{\otimes }_{A}{\Omega }^{•+1}A$\nabla : M\otimes_A\Omega^\bullet A \to M\otimes_A\Omega^{\bullet+1} A

such that for any homogenous element $\omega \in {\Omega }^{k}A$ and an element $\chi \in \Omega A$

$\nabla \left(\omega \chi \right)=\nabla \left(\omega \right)\chi +\left(-1{\right)}^{k}\omega \nabla \left(\chi \right)$\nabla (\omega\chi) = \nabla(\omega)\chi + (-1)^k\omega\nabla(\chi)

The curvature ${F}_{\nabla }$ of the connection $\nabla$ is the restriction of the connection squared to $M$:

$\nabla \circ \nabla {\mid }_{M}:M\to M{\otimes }_{A}{\Omega }^{2}A.$\nabla\circ\nabla|_M : M\to M\otimes_A\Omega^2 A.

A connection is flat (or integrable) iff its curvature vanishes.

See also connection in noncommutative geometry as some versions are close to this approach. A past query on history of the notion of connection for a dga is archived here.

Revised on September 6, 2011 21:24:12 by Zoran Škoda (161.53.130.104)