On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology : Notas de Matemática, 59

In section 1. the notion of (strong) nil- homotopy is introduced, and its first properties are proved. Section 2 is devoted to the interpretation of quasi-free pro-algebras as cofibrant objects with respect to the setting of the previous section (Theorem 2.1). The notion of weak nil-homotopy is intr...

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Autor principal: Cortiñas, Guillermo Horacio
Formato: Publicacion seriada
Lenguaje:Inglés
Publicado: 1997
Materias:
Matemática
Acceso en línea:http://sedici.unlp.edu.ar/handle/10915/171047
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SEDICI (UNLP) de Universidad Nacional de La Plata
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spelling I19-R120-10915-1710472024-10-02T20:11:35Z http://sedici.unlp.edu.ar/handle/10915/171047 On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology : Notas de Matemática, 59 Cortiñas, Guillermo Horacio 1997 2024-10-02T16:41:30Z en Matemática In section 1. the notion of (strong) nil- homotopy is introduced, and its first properties are proved. Section 2 is devoted to the interpretation of quasi-free pro-algebras as cofibrant objects with respect to the setting of the previous section (Theorem 2.1). The notion of weak nil-homotopy is introduced in section 3, where the existence of the localized category Def lPA is proved (Theorem 1. A Closed Model Category Analogy. 1.0 We consider associative, non-necessarily unital algebras over a fixed ground field k. We write A and V for the categories of algebras and vector spaces and PA and VV for the corresponding pro-categories. As in [CQ3] we consider only countably indexed pro-objects. A map f E PA(A.B) is called a fibration if it admits a right inverse as a map of pro-vector spaces, i.e. there exists s E PV(B.A) such that fs = 1. Fibrations are denoted by a double headed arrow By a (nil-) deformation (-») of a pro-algebra 3.2) . Section 2 is devoted to the comparison between our notion of nil-homotopv and the usual, polynomial homotopy. We prove that the localization at the union of the classes of nil-deformations and graded deformations exists and can be calculated as a homotopy category (Theorem 4.1). Section 5 deals with the formalization of the derived functor analogy of [CQ2]. We establish sufficient conditions for the existence of left derived functors (Theorem 5.2) and prove that, in characteristic zero, these conditions are met by the de Rham supercoinplex functor A θ- XA of Cuntz-Quillen (Corollary 5.4). In section 6 we compute the derived functor of the rational A’-theorv of rational pro-algebras. (Theorem 6.2) and of the negative cyclic homology of pro-algebras over any field (Corollary 6.9). Material digitalizado en SEDICI gracias a la colaboración de la Biblioteca del Departamento de Matemática de la Facultad de Ciencias Exactas (UNLP). Facultad de Ciencias Exactas Publicacion seriada Publicacion seriada http://creativecommons.org/licenses/by-nc-sa/4.0/ Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) application/pdf
institution Universidad Nacional de La Plata
institution_str I-19
repository_str R-120
collection SEDICI (UNLP)
language Inglés
topic Matemática
spellingShingle Matemática
Cortiñas, Guillermo Horacio
On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology : Notas de Matemática, 59
topic_facet Matemática
description In section 1. the notion of (strong) nil- homotopy is introduced, and its first properties are proved. Section 2 is devoted to the interpretation of quasi-free pro-algebras as cofibrant objects with respect to the setting of the previous section (Theorem 2.1). The notion of weak nil-homotopy is introduced in section 3, where the existence of the localized category Def lPA is proved (Theorem 1. A Closed Model Category Analogy. 1.0 We consider associative, non-necessarily unital algebras over a fixed ground field k. We write A and V for the categories of algebras and vector spaces and PA and VV for the corresponding pro-categories. As in [CQ3] we consider only countably indexed pro-objects. A map f E PA(A.B) is called a fibration if it admits a right inverse as a map of pro-vector spaces, i.e. there exists s E PV(B.A) such that fs = 1. Fibrations are denoted by a double headed arrow By a (nil-) deformation (-») of a pro-algebra 3.2) . Section 2 is devoted to the comparison between our notion of nil-homotopv and the usual, polynomial homotopy. We prove that the localization at the union of the classes of nil-deformations and graded deformations exists and can be calculated as a homotopy category (Theorem 4.1). Section 5 deals with the formalization of the derived functor analogy of [CQ2]. We establish sufficient conditions for the existence of left derived functors (Theorem 5.2) and prove that, in characteristic zero, these conditions are met by the de Rham supercoinplex functor A θ- XA of Cuntz-Quillen (Corollary 5.4). In section 6 we compute the derived functor of the rational A’-theorv of rational pro-algebras. (Theorem 6.2) and of the negative cyclic homology of pro-algebras over any field (Corollary 6.9).
format Publicacion seriada
Publicacion seriada
author Cortiñas, Guillermo Horacio
author_facet Cortiñas, Guillermo Horacio
author_sort Cortiñas, Guillermo Horacio
title On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology : Notas de Matemática, 59
title_short On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology : Notas de Matemática, 59
title_full On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology : Notas de Matemática, 59
title_fullStr On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology : Notas de Matemática, 59
title_full_unstemmed On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology : Notas de Matemática, 59
title_sort on the derived functor analogy in the cuntz-quillen framework for cyclic homology : notas de matemática, 59
publishDate 1997
url http://sedici.unlp.edu.ar/handle/10915/171047
work_keys_str_mv AT cortinasguillermohoracio onthederivedfunctoranalogyinthecuntzquillenframeworkforcyclichomologynotasdematematica59
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