connection for a differential graded algebra

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

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

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

$\nabla (\omega\chi) = \nabla(\omega)\chi + (-1)^k\omega\nabla(\chi)$

The curvature $F_\nabla$ of the connection $\nabla$ is the restriction of the connection squared to $M$:

$\nabla\circ\nabla|_M : M\to M\otimes_A\Omega^2 A.$

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)