Comparison Maps for Relatively Free Resolutions

Let Λ be a commutative ring, A an augmented differential graded algebra over Λ (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.