Artículo
Abstract:
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. © 2003 Elsevier B.V. All rights reserved.
Registro:
| Documento: | Artículo |
| Título: | Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence |
| Autor: | Castiglioni, J.L.; Cortiñas, G. |
| Filiación: | Departamento de Matemática, Facultad de Cs. Exactas, Calle 50 y 115, 1900 La Plata, Argentina Departamento de Matemática, Pabellón 1, Buenos Aires 1428, Argentina |
| Año: | 2004 |
| Volumen: | 191 |
| Número: | 1-2 |
| Página de inicio: | 119 |
| Página de fin: | 142 |
| DOI: | http://dx.doi.org/10.1016/j.jpaa.2003.11.009 |
| Título revista: | Journal of Pure and Applied Algebra |
| Título revista abreviado: | J. Pure Appl. Algebra |
| ISSN: | 00224049 |
| CODEN: | JPAAA |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00224049_v191_n1-2_p119_Castiglioni |
Referencias:
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Citas:
---------- APA ----------
Castiglioni, J.L. & Cortiñas, G. (2004) . Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence. Journal of Pure and Applied Algebra, 191(1-2), 119-142.http://dx.doi.org/10.1016/j.jpaa.2003.11.009
---------- CHICAGO ----------
Castiglioni, J.L., Cortiñas, G. "Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence" . Journal of Pure and Applied Algebra 191, no. 1-2 (2004) : 119-142.http://dx.doi.org/10.1016/j.jpaa.2003.11.009
---------- MLA ----------
Castiglioni, J.L., Cortiñas, G. "Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence" . Journal of Pure and Applied Algebra, vol. 191, no. 1-2, 2004, pp. 119-142.http://dx.doi.org/10.1016/j.jpaa.2003.11.009
---------- VANCOUVER ----------
Castiglioni, J.L., Cortiñas, G. Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence. J. Pure Appl. Algebra. 2004;191(1-2):119-142.http://dx.doi.org/10.1016/j.jpaa.2003.11.009
