connection for a differential graded algebra

Given a semi-free differential graded algebra ${\Omega}^{\u2022}A$ over $k$-algebra $A$, a **connection** in $A$-module $M$ is a $k$-linear map

$$\nabla :M{\otimes}_{A}{\Omega}^{\u2022}A\to M{\otimes}_{A}{\Omega}^{\u2022+1}A$$

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 $M$:

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

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)