## (1999)

### Abstract

Analyzing the transference of the coalgebra structure on the homology of CDGAs

### Citations

406 | An introduction to homological algebra, Cambridge Studies - Weibel - 1994 |

66 | Perturbation theory in differential homological algebra
- Gugenheim, Lambe
- 1989
(Show Context)
Citation Context ...DGAs. Our main technique is the use of Homological Perturbation Theory which is often used to replace given chain complexes by homotopic, smaller and more readily computable chain complexes ([7], [8],=-=[9]-=-). The Basic Perturbation Lemma (BPL) states that given a contraction (f; g; OE) : (N; dN) ) (M; dM) (1) 1sof chain complexes (with f g = 1M and 1N \Gammasgf = dNOE + OEdN ) and a perturbation ffi of ... |

50 |
Small models for chain algebras
- Huebschmann, Kadeishvili
- 1991
(Show Context)
Citation Context ...h that H\Lambda (M; dM + dffi) = H\Lambda (N; dN + ffi). Perturbation results about preservation of structures (DG-algebras, DG-coalgebras, Lie algebras, : : : ) have been largely considered ([9] and =-=[10]-=-). The technique for obtaining these results is to determine under what conditions the BPL allows the preservation of the data structures. In the case of algebra contractions, the study of the preserv... |

41 |
On the chain complex of a fibration
- Gugenheim
- 1972
(Show Context)
Citation Context ...logy of CDGAs. Our main technique is the use of Homological Perturbation Theory which is often used to replace given chain complexes by homotopic, smaller and more readily computable chain complexes (=-=[7]-=-, [8],[9]). The Basic Perturbation Lemma (BPL) states that given a contraction (f; g; OE) : (N; dN) ) (M; dM) (1) 1sof chain complexes (with f g = 1M and 1N \Gammasgf = dNOE + OEdN ) and a perturbatio... |

19 |
Classics in Mathematics
- Homology
- 1995
(Show Context)
Citation Context ...nly known that every commutative DGA A "factors" up to homotopy equivalence into a twisted tensor product of exterior and polynomial algebras endowed with a differential-derivation (see, for example, =-=[12]-=-); in the sense that there exists an homomorphism connecting both structures, which induces an isomorphism in homology. This last TTP is called a free model for A. In fact, the object we start from is... |

18 | Homological perturbation theory, Hochschild homology, and formal groups. Deformation theory and quantum groups with applications to mathematical physics - Lambe - 1990 |

8 |
Homologie effective des espaces de lacets it'er'es: un logiciel. Th`ese de Math. de l'Universit'e
- Rubio
- 1991
(Show Context)
Citation Context ...nly set explicit formulae for the projection and inclusion morphisms, an explicit formula of the recursive definition of the homotopy operator given by Eilenberg and Mac Lane in [5] is established in =-=[14]-=- (see also [13] for a proof). Given a tensor product \Omega i2I Ai of augmented CDGA-algebras, a contraction from_ B(\Omega i2I Ai) to \Omega i2I _B(Ai) is easily determined, by applying C _B\Omegasse... |

5 |
On the Theory and
- Gugenheim, May
- 1974
(Show Context)
Citation Context ...of CDGAs. Our main technique is the use of Homological Perturbation Theory which is often used to replace given chain complexes by homotopic, smaller and more readily computable chain complexes ([7], =-=[8]-=-,[9]). The Basic Perturbation Lemma (BPL) states that given a contraction (f; g; OE) : (N; dN) ) (M; dM) (1) 1sof chain complexes (with f g = 1M and 1N \Gammasgf = dNOE + OEdN ) and a perturbation ffi... |

3 | Homological Perturbation Theory and computability of Hochschild and cyclic homologies of CDGAs
- Alvarez, Silva
- 1997
(Show Context)
Citation Context ...euler.fie.us.es, real@cica.es, silva@cica.es May 6, 1999 Abstract Our motivation here is to analyze the A1-coalgebra structure (\Delta 2; \Delta 3; : : :) of the small homological model H obtained in =-=[2]-=-, for any commutative differential graded algebra (or briefly CDGA) A. More precisely, making use of the facts that H is a CDGA and that the morphism \Delta 2 : H ! H \OmegasH satisfies a compatibilit... |

3 |
On p-minimal homological models of TTPs of elementary complexes localized at a prime
- Armario, Real, et al.
- 1999
(Show Context)
Citation Context ...amined in terms of semifull algebra contractions in [2], and "small" p-local homological models of reduced bar constructions of twisted tensor product of Cartan's elementary complexes are obtained in =-=[4]-=-, due to the fact that all the contractions appearing there, are of this type. These examples will be our starting point in this paper, being the transference of the coalgebra structure in the previou... |

3 |
Homological Perturbation Theory and Associativity. Preprint n
- Real
- 1996
(Show Context)
Citation Context ...t formulae for the projection and inclusion morphisms, an explicit formula of the recursive definition of the homotopy operator given by Eilenberg and Mac Lane in [5] is established in [14] (see also =-=[13]-=- for a proof). Given a tensor product \Omega i2I Ai of augmented CDGA-algebras, a contraction from_ B(\Omega i2I Ai) to \Omega i2I _B(Ai) is easily determined, by applying C _B\Omegasseveral times in ... |

2 |
On the groups H(ss
- Eilenberg, Lane
- 1953
(Show Context)
Citation Context ...lenberg and Mac Lane only set explicit formulae for the projection and inclusion morphisms, an explicit formula of the recursive definition of the homotopy operator given by Eilenberg and Mac Lane in =-=[5]-=- is established in [14] (see also [13] for a proof). Given a tensor product \Omega i2I Ai of augmented CDGA-algebras, a contraction from_ B(\Omega i2I Ai) to \Omega i2I _B(Ai) is easily determined, by... |

1 | Gonz'alez-D'iaz and P.Real. Algorithms in Algebraic Topology and Homological Algebra: the problem of the complexity - 'Alvarez, Armario, et al. - 1998 |

1 |
Computability of the algebra homology of CDGAs
- 'Alvarez, Armario, et al.
(Show Context)
Citation Context ... whether \Delta 2 is coassociative or not. At first, it would be necessary to evaluate \Delta 2 over all the module generators of HBA. Nevertheless, since the projection f is a morphism of CDGAs (see =-=[3]-=-), it is easy to prove that \Delta 2 satisfies the following condition: \Delta 2 ffi _ = (_ \Omegas_)(1 \OmegasT \Omegas1)(\Delta 2 \Omegas\Delta 2); where _ : HBA \OmegasHBA ! HBA is the product of t... |