2 research outputs found
Comparison maps for relatively free resolutions
Let Ī be a commutative ring, A an augmented differential graded algebra over Ī (briefly, DGA-algebra) and X be a relatively free resolution of Ī over A. The standard bar resolution of Ī over A, denoted by B(A), provides an example of a resolution of this kind. The comparison theorem gives inductive formulae f : B(A)āX and g : XāB(A) termed comparison maps. In case that fg=1 X and A is connected, we show that X is endowed a A āāā-tensor product structure. In case that A is in addition commutative then (X,Ī¼ X ) is shown to be a commutative DGA-algebra with the product Ī¼ X =f*(gāg) (* 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