Comparison maps for relatively free resolutions

Abstract

Let Λ be a, commutative ring, A an augmented differential graded algebra over A (briefly, DGA-algebra) and X be a relatively free resolution of Λ over A. The standard bar resolution of Λ over A, denoted by B(A), provides an example of a resolution of this kind. The comparison, theorem gives inductive formulae f: B(A) → X and g: X → B(A) termed comparison maps. In case that fg = 1x and A is connected, we show that X is endowed a A∞-tensor product structure. In case that A is in addition commutative then (X, μx) is shown to be a commutative DGA-algebra with the product μx = f * (g ⊗g) (* is the shuffle product in B(A)). Furthermore, f and g are algebra maps. We give an example in order to illustrate the main results of this paper. © Springer-Verlag Berlin Heidelberg 2006.