Abstract
Our main result here is a rational computation of the homology of the adjoint action of the infinite general linear group of an arbitrary ring. Before stating the result we establish some notation and conventions. Rings are associative and with unit. If A is a ring then GL(A)= Uk?0GLk(A) is its infinite general linear group. An A-bimodule is an abelian group B which is both a left A-module and a right A-module and satisfies (alb)a2 = al(ba2) for ai eA, be B (for example B = A). If B is an A-bimodule, then M(B) = Uk?oMk(B) is the infinite additive group of matrices with entries in B. Conjugation defines an action (the adjoint action) of GL(A) on M(B). Note that an A ? Q-bimodule is just an A-bimodule which is also a rational vector space. If B is an A ? Q-bimodule, then HJ(A C) Q; B) denotes the Hochschild homology of A ? Q with coefficients in B. Our main result (it appears in slightly more detailed form as Theorem V.3) is:
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