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Comparison Maps for Relatively Free Resolutions

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Computer Algebra in Scientific Computing (CASC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4194))

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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.

This work was partially supported by the PAICYT research project FQM–296 from Junta de Andalucía (Spain).

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Authors and Affiliations

  1. Depto. Matemática Aplicada I, E.T.S.I. Informática, Universidad de Sevilla, Avda. Reina Mercedes, S.N., 41012, Sevilla, (Spain)

    V. Álvarez, J. A. Armario, M. D. Frau & P. Real

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  1. V. Álvarez
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  2. J. A. Armario
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  3. M. D. Frau
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  4. P. Real
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Editors and Affiliations

  1. Institute of Informatics, Technische Universität München, Boltzmannstr. 3, 85748, Garching, Germany

    Victor G. Ganzha  & Ernst W. Mayr  & 

  2. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Álvarez, V., Armario, J.A., Frau, M.D., Real, P. (2006). Comparison Maps for Relatively Free Resolutions. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2006. Lecture Notes in Computer Science, vol 4194. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11870814_1

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  • DOI: https://doi.org/10.1007/11870814_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-45182-2

  • Online ISBN: 978-3-540-45195-2

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