Higher homotopies and Golod rings

We study the homological algebra of an R = Q/I module M using A-infinity structures on Q-projective resolutions of R and M. We use these higher homotopies to construct an R-projective bar resolution of M, Q-projective resolutions for all R-syzygies of M, and describe the differentials in the Avramov spectral sequence for M. These techniques apply particularly well to Golod modules over local rings. We characterize R-modules that are Golod over Q as those with minimal A-infinity structures. This gives a construction of the minimal resolution of every module over a Golod ring, and it also follows that if the inequality traditionally used to define Golod modules is an equality in the first dim Q+1 degrees, then the module is Golod, where no bound was previously known. We also relate A-infinity structures on resolutions to Avramov's obstructions to the existence of a dg-module structure. Along the way we give new, shorter, proofs of several classical results about Golod modules.