Comparison Maps for Relatively Free Resolutions

  • V. Álvarez
  • J. A. Armario
  • M. D. Frau
  • P. Real
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4194)


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 : XB(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*(gg) (* 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.


Comparison Theorem Algebra Structure Free Resolution Perturbation Data Homotopy Operator 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • V. Álvarez
    • 1
  • J. A. Armario
    • 1
  • M. D. Frau
    • 1
  • P. Real
    • 1
  1. 1.Depto. Matemática Aplicada I, E.T.S.I. InformáticaUniversidad de SevillaSevilla(Spain)

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