The Homotopy Groups of the Simplicial Mapping Space between Algebras
Documenta mathematica, Tome 24 (2019), pp. 251-270
Let l be a commutative ring with unit. To every pair of l-algebras A and B one can associate a simplicial set Hom(A,BΔ) so that π0Hom(A,BΔ) equals the set of polynomial homotopy classes of morphisms from A to B. We prove that πnHom(A,BΔ) is the set of homotopy classes of morphisms from A to B∙Sn, where B∙Sn is the ind-algebra of polynomials on the n-dimensional cube with coefficients in B vanishing at the boundary of the cube. This is a generalization to arbitrary dimensions of a theorem of Cortiñas-Thom, which addresses the cases n≤1. As an application we give a simplified proof of a theorem of Garkusha that computes the homotopy groups of his matrix-unstable algebraic KK-theory space in terms of polynomial homotopy classes of morphisms.
Classification :
19K35, 55Q52
Mots-clés : homotopy theory of algebras, bivariant algebraic K-theory
Mots-clés : homotopy theory of algebras, bivariant algebraic K-theory
@article{10_4171_dm_680,
author = {Emanuel Dar{\'\i}o Rodr{\'\i}guez Cirone},
title = {The {Homotopy} {Groups} of the {Simplicial} {Mapping} {Space} between {Algebras}},
journal = {Documenta mathematica},
pages = {251--270},
year = {2019},
volume = {24},
doi = {10.4171/dm/680},
url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/680/}
}
Emanuel Darío Rodríguez Cirone. The Homotopy Groups of the Simplicial Mapping Space between Algebras. Documenta mathematica, Tome 24 (2019), pp. 251-270. doi: 10.4171/dm/680
Cité par Sources :