connection for a differential graded algebra

Given a semi-free differential graded algebra Ω A over k-algebra A, a connection in A-module M is a k-linear map

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

such that for any homogenous element ωΩ kA and an element χΩA

(ωχ)=(ω)χ+(1) kω(χ)\nabla (\omega\chi) = \nabla(\omega)\chi + (-1)^k\omega\nabla(\chi)

The curvature F of the connection is the restriction of the connection squared to M:

M:MM AΩ 2A.\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 (