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Derived localisation of algebras and modules

Research output: Contribution to journalJournal article

Published
<mark>Journal publication date</mark>13/04/2018
<mark>Journal</mark>Advances in Mathematics
Volume328
Number of pages68
Pages (from-to)555-622
<mark>State</mark>Published
Early online date8/02/18
<mark>Original language</mark>English

Abstract

For any dg algebra A, not necessarily commutative, and a subset S in H(A)H(A), the homology of A , we construct its derived localisation LS(A)LS(A) together with a map A→LS(A)A→LS(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, LS(A)LS(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(LS(A))H(LS(A)) and LS(H(A))LS(H(A)); this spectral sequence collapses when, e.g. S is an Ore set or when A is a free ring.

We prove that LS(A)LS(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.

Bibliographic note

This is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics, 328, 2018 DOI: 10.1016/j.aim.2018.02.004