Homological perturbation theory, Hochschild homology, and formal groups

Lambe, Larry A.

doi:10.1090/conm/134/1187288

Cited by 17 publications

(30 citation statements)

References 20 publications

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“…An important example of relatively free and contractile resolution of Λ over A is the bar resolution (B(A), s, d) (or B(A)) [16,13]. More concretely, B(A) is the twisted tensor product A ⊗ θB (A), where the twisting cochain θ is given by…”

Section: The Canonical Comparison Contraction a Necessary And Sufficmentioning

confidence: 99%

“…These formulae are crucial for the work in [13,14]. Let us observe that the morphisms g and φ satisfy that…”

Section: The Canonical Comparison Contraction a Necessary And Sufficmentioning

confidence: 99%

“…An explicit formula for φ is given in [13,14] which increases the simplicial degree inB(A) by one and φ 0 = 0. Therefore, θ φ|B (A) = 0 since θ:B(A) → A is the universal twisting cochain.…”

Section: Theoremmentioning

confidence: 99%

“…Using this contraction and some homological perturbation tools, perturbations of this resolution can be computed and the perturbed resolutions can be used to make complete effective calculations where previously only partial or indirect results were obtainable. This idea has been exploited in a series of papers by Lambe [12][13][14] and provides an algorithm for computing resolutions which split off of the bar construction. This algorithm has been extended to a more general context in [11].…”

Section: Introductionmentioning

confidence: 99%

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Comparison Maps for Relatively Free Resolutions

Álvarez

Armario

Frau

et al. 2006

Computer Algebra in Scientific Computing

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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 f g = 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.

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Section: The Canonical Comparison Contraction a Necessary And Sufficmentioning

confidence: 99%

“…These formulae are crucial for the work in [13,14]. Let us observe that the morphisms g and φ satisfy that…”

Section: The Canonical Comparison Contraction a Necessary And Sufficmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Comparison Maps for Relatively Free Resolutions

Álvarez

Armario

Frau

et al. 2006

Computer Algebra in Scientific Computing

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show abstract

“…This is done using homological perturbation theory [4], [10], [1], [11], [19], [21] and Gröbner basis theory [8], [3].…”

Section: Introductionmentioning

confidence: 99%

On constructing resolutions over the polynomial

Johansson

Lambe

Sköldberg

2002

Homology, Homotopy and Applications

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A Mathematica Notebook for Computing the Homology of Iterated Products of Groups

Álvarez

Armario

Frau

et al. 2006

Lecture Notes in Computer Science

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Abstract. Let G be a group which admits the structure of an iterated product of central extensions and semidirect products of abelian groups Gi (both finite and infinite). We describe a Mathematica 4.0 notebook for computing the homology of G, in terms of some homological models for the factor groups Gi and the products involved. Computational results provided by our program have allowed the simplification of some of the formulae involved in the calculation of Hn (G). Consequently the efficiency of the method has been improved as well. We include some executions and examples.

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Homological perturbation theory, Hochschild homology, and formal groups

Cited by 17 publications

References 20 publications

Comparison Maps for Relatively Free Resolutions

Comparison Maps for Relatively Free Resolutions

On constructing resolutions over the polynomial

A Mathematica Notebook for Computing the Homology of Iterated Products of Groups

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