connection for a differential graded algebra

Given a semi-free differential graded algebra Ω A\Omega^\bullet A over kk-algebra AA, a connection in AA-module MM is a kk-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\omega\in\Omega^k A and an element χΩA\chi\in\Omega A

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

The curvature F F_\nabla of the connection \nabla is the restriction of the connection squared to MM:

| 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.

Last revised on September 6, 2011 at 21:24:12. See the history of this page for a list of all contributions to it.