Derived localisation of algebras and modules

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 LS(A) together with a map 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) 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)) and 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) 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.