Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence
- Autores
- Castiglioni, José Luis; Cortiñas, Guillermo Horacio
- Año de publicación
- 2004
- Idioma
- inglés
- Tipo de recurso
- artículo
- Estado
- versión publicada
- Descripción
- The (dual) Dold-Kan correspondence says that there is an equivalence of categories K: Ch≥0→AbΔ between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show that the restriction of K to DG-rings can be equipped with an associative product and that the resulting functor DGR*→RingsΔ, although not itself an equivalence, does induce one at the level of homotopy categories. In other words both DGR* and RingsΔ are Quillen closed model categories and the total left derived functor of K is an equivalence: LK: Ho DGR* Ho RingsΔ. The dual of this result for chain DG and simplicial rings was obtained independently by Schwede and Shipley, Algebraic and Geometric Topology 3 (2003) 287, through different methods. Our proof is based on a functor Q:DGR*→RingsΔ, naturally homotopy equivalent to K, and which preserves the closed model structure. It also has other interesting applications. For example, we use Q to prove a noncommutative version of the Hochschild-Kostant-Rosenberg and Loday-Quillen theorems. Our version applies to the cyclic module [n] ∐nRS that arises from a homomorphism R→S of not necessarily commutative rings, using the coproduct ∐R of associative R-algebras. As another application of the properties of Q, we obtain a simple, braid-free description of a product on the tensor power S⊗Rn originally defined by Nuss K-theory 12 (1997) 23, using braids.
Facultad de Ciencias Exactas - Materia
-
Matemática
Álgebra - Nivel de accesibilidad
- acceso abierto
- Condiciones de uso
- http://creativecommons.org/licenses/by-nc-sa/4.0/
- Repositorio
- Institución
- Universidad Nacional de La Plata
- OAI Identificador
- oai:sedici.unlp.edu.ar:10915/83543
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Cosimplicial versus DG-rings: A version of the Dold-Kan correspondenceCastiglioni, José LuisCortiñas, Guillermo HoracioMatemáticaÁlgebraThe (dual) Dold-Kan correspondence says that there is an equivalence of categories K: Ch≥0→AbΔ between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show that the restriction of K to DG-rings can be equipped with an associative product and that the resulting functor DGR*→RingsΔ, although not itself an equivalence, does induce one at the level of homotopy categories. In other words both DGR* and RingsΔ are Quillen closed model categories and the total left derived functor of K is an equivalence: LK: Ho DGR* Ho RingsΔ. The dual of this result for chain DG and simplicial rings was obtained independently by Schwede and Shipley, Algebraic and Geometric Topology 3 (2003) 287, through different methods. Our proof is based on a functor Q:DGR*→RingsΔ, naturally homotopy equivalent to K, and which preserves the closed model structure. It also has other interesting applications. For example, we use Q to prove a noncommutative version of the Hochschild-Kostant-Rosenberg and Loday-Quillen theorems. Our version applies to the cyclic module [n] ∐nRS that arises from a homomorphism R→S of not necessarily commutative rings, using the coproduct ∐R of associative R-algebras. As another application of the properties of Q, we obtain a simple, braid-free description of a product on the tensor power S⊗Rn originally defined by Nuss K-theory 12 (1997) 23, using braids.Facultad de Ciencias Exactas2004info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionArticulohttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdf119-142http://sedici.unlp.edu.ar/handle/10915/83543enginfo:eu-repo/semantics/altIdentifier/issn/0022-4049info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jpaa.2003.11.009info:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by-nc-sa/4.0/Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)reponame:SEDICI (UNLP)instname:Universidad Nacional de La Platainstacron:UNLP2025-10-15T11:07:43Zoai:sedici.unlp.edu.ar:10915/83543Institucionalhttp://sedici.unlp.edu.ar/Universidad públicaNo correspondehttp://sedici.unlp.edu.ar/oai/snrdalira@sedici.unlp.edu.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:13292025-10-15 11:07:43.372SEDICI (UNLP) - Universidad Nacional de La Platafalse |
dc.title.none.fl_str_mv |
Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence |
title |
Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence |
spellingShingle |
Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence Castiglioni, José Luis Matemática Álgebra |
title_short |
Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence |
title_full |
Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence |
title_fullStr |
Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence |
title_full_unstemmed |
Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence |
title_sort |
Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence |
dc.creator.none.fl_str_mv |
Castiglioni, José Luis Cortiñas, Guillermo Horacio |
author |
Castiglioni, José Luis |
author_facet |
Castiglioni, José Luis Cortiñas, Guillermo Horacio |
author_role |
author |
author2 |
Cortiñas, Guillermo Horacio |
author2_role |
author |
dc.subject.none.fl_str_mv |
Matemática Álgebra |
topic |
Matemática Álgebra |
dc.description.none.fl_txt_mv |
The (dual) Dold-Kan correspondence says that there is an equivalence of categories K: Ch≥0→AbΔ between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show that the restriction of K to DG-rings can be equipped with an associative product and that the resulting functor DGR*→RingsΔ, although not itself an equivalence, does induce one at the level of homotopy categories. In other words both DGR* and RingsΔ are Quillen closed model categories and the total left derived functor of K is an equivalence: LK: Ho DGR* Ho RingsΔ. The dual of this result for chain DG and simplicial rings was obtained independently by Schwede and Shipley, Algebraic and Geometric Topology 3 (2003) 287, through different methods. Our proof is based on a functor Q:DGR*→RingsΔ, naturally homotopy equivalent to K, and which preserves the closed model structure. It also has other interesting applications. For example, we use Q to prove a noncommutative version of the Hochschild-Kostant-Rosenberg and Loday-Quillen theorems. Our version applies to the cyclic module [n] ∐nRS that arises from a homomorphism R→S of not necessarily commutative rings, using the coproduct ∐R of associative R-algebras. As another application of the properties of Q, we obtain a simple, braid-free description of a product on the tensor power S⊗Rn originally defined by Nuss K-theory 12 (1997) 23, using braids. Facultad de Ciencias Exactas |
description |
The (dual) Dold-Kan correspondence says that there is an equivalence of categories K: Ch≥0→AbΔ between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show that the restriction of K to DG-rings can be equipped with an associative product and that the resulting functor DGR*→RingsΔ, although not itself an equivalence, does induce one at the level of homotopy categories. In other words both DGR* and RingsΔ are Quillen closed model categories and the total left derived functor of K is an equivalence: LK: Ho DGR* Ho RingsΔ. The dual of this result for chain DG and simplicial rings was obtained independently by Schwede and Shipley, Algebraic and Geometric Topology 3 (2003) 287, through different methods. Our proof is based on a functor Q:DGR*→RingsΔ, naturally homotopy equivalent to K, and which preserves the closed model structure. It also has other interesting applications. For example, we use Q to prove a noncommutative version of the Hochschild-Kostant-Rosenberg and Loday-Quillen theorems. Our version applies to the cyclic module [n] ∐nRS that arises from a homomorphism R→S of not necessarily commutative rings, using the coproduct ∐R of associative R-algebras. As another application of the properties of Q, we obtain a simple, braid-free description of a product on the tensor power S⊗Rn originally defined by Nuss K-theory 12 (1997) 23, using braids. |
publishDate |
2004 |
dc.date.none.fl_str_mv |
2004 |
dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion Articulo http://purl.org/coar/resource_type/c_6501 info:ar-repo/semantics/articulo |
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article |
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http://sedici.unlp.edu.ar/handle/10915/83543 |
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eng |
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eng |
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info:eu-repo/semantics/altIdentifier/issn/0022-4049 info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jpaa.2003.11.009 |
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info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by-nc-sa/4.0/ Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) |
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http://creativecommons.org/licenses/by-nc-sa/4.0/ Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) |
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