Abstract

For any dg algebra A , not necessarily commutative, and a subset S in H ( A ) , the homology of A , we construct its derived localisation L S ( A ) together with a map A → L S ( A ) , well-defined in the homotopy category of dg algebras, which possesses a universal property, similar to that of the ordinary localisation, but formulated in homotopy invariant terms. Even if A is an ordinary ring, L S ( A ) may have non-trivial homology. Unlike the commutative case, the localisation functor does not commute, in general, with homology but instead there is a spectral sequence relating H ( L S ( A )) and L S ( H ( A )) ; this spectral sequence collapses when, e.g. S is an Ore set or when A is a free ring. We prove that L S ( A ) could also be regarded as a Bousfield localisation of A viewed as a left or right dg module over itself. Combined with the results of Dwyer–Kan on simplicial localisation, this leads to a simple and conceptual proof of the topological group completion theorem. Further applications include algebraic K –theory, cyclic and Hochschild homology, strictification of homotopy unital algebras, idempotent ideals, the stable homology of various mapping class groups and Kontsevich’s graph homology

Publication Type:	Article
Additional Information:	© 2018 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Publisher Keywords:	derived localisation, dg algebra, Ore set, group completion.
Departments:	School of Science & Technology > Mathematics

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