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Comparison maps for relatively free resolutions


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Author: Álvarez Solano, Víctor
Armario Sampalo, José Andrés
Frau García, María Dolores
Real Jurado, Pedro
Department: Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII)
Date: 2006
Published in: Computer Algebra in Scientific Computing, Lecture Notes in Computer Science, Vol.4194, p. 1-22
Document type: Chapter of Book
Abstract: 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=1 X 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,μ X ) is shown to be a commutative DGA-algebra with the product μ X =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.
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