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Computing "Small" 1Homological Models for Commutative Differential Graded Algebras
"... We use homological perturbation machinery specific for the algebra category [13] to give an algorithm for computing the differential structure of a small 1homological model for commutative differential graded algebras (briefly, CDGAs). The complexity of the procedure is studied and a computer packa ..."
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We use homological perturbation machinery specific for the algebra category [13] to give an algorithm for computing the differential structure of a small 1homological model for commutative differential graded algebras (briefly, CDGAs). The complexity of the procedure is studied and a computer package in Mathematica is described for determining such models.
An Algorithm for Computing Cocyclic Matrices Developed Over Some Semidirect Products
"... An algorithm for calculating a set of generators of representative 2cocycles on semidirect product of finite abelian groups is constructed, in light of the theory over cocyclic matrices developed by Horadam and de Launey in [7, 8]. The method involves some homological perturbation techniques [3, 1] ..."
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An algorithm for calculating a set of generators of representative 2cocycles on semidirect product of finite abelian groups is constructed, in light of the theory over cocyclic matrices developed by Horadam and de Launey in [7, 8]. The method involves some homological perturbation techniques [3, 1], in the homological correspondent to the work which Grabmeier and Lambe described in [12] from the viewpoint of cohomology. Examples of explicit computations over all dihedral groups D4t are given, with aid of Mathematica. 1
Comparison maps for relatively free resolutions ⋆
"... Abstract. Let Λ be a commutative ring, A an augmented differential graded algebra over Λ (briefly, DGAalgebra) 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 ..."
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Abstract. Let Λ be a commutative ring, A an augmented differential graded algebra over Λ (briefly, DGAalgebra) 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 = 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 DGAalgebra 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. 1
Algorithms In Algebraic Topology And Homological Algebra: Problem Of Complexity
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