Summary. Let R be a commutative finite dimensional noetherian ring or, more generally, an associative ring which satisfies one of Bass' stable range conditions. We describe a modified version of H. Maazen's work [18], yielding stability for the homology of linear groups over R. Applying W.G. Dwyer's arguments (cf.[9]) we also get stability for homology with twisted coefficients. For example, H2(GL,(R), R n) takes on a stable value when n becomes large. w 1. Introduction 1.1. Our motivation for this work has been to prove stability for algebraic Ktheory in BGL + context. Thanks to the recent work of Dwyer we actually get much more general statements. These imply a result which seems to be of interest to geometric topologists. Namely, we find that the twisted homology groups Hk (GL,(R), p,), considered by Dwyer in [9], stabilize with respect to n not only when R is a PID, but also when R is the group ring Z [n] of a finite group n. This fits in with work of W.G. Dwyer, Wu-Chung Hsiang and R.E. Staffeldt on Waldhausen's rational algebraic K-groups of a space. Let us remind the reader what sort of families {Pn} are considered byDwyer, leaving out all technicalities and using some suggestive but unexplained terminology. A basic example is the family 2={~,}, where ;t, denotes (the standard representation of GL,(R) in) the right R-module R n of column vectors of length n over R. This system 2 grows linearly with n. Note that the difference between 2,+ 1 and 7. is equal to R for all n, so that the system of differences is constant in this case. More generally Dwyer considers systems that grow polynomially with n, such as the system # = {#,}, where/~n denotes (the representation by conjugation of GL,(R) in) the space of n by n matrices over R. The system/~ grows quadratically with n, which can be rephrased by saying that its system of third iterated differences is zero, while its system of second iterated differences is not zero. (To make sense of all this, one has to add more structure 0020-9910/80/0060/0269/$05.40

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O. IntroductionStable range conditions on a ring R were devised by H. Bass in order to determine values of n for which every matrix in GL,(R) can be row reduced (by addition operations with coefficients from R) to a matrix with the same last row and column as the identity matrix 1,. In order to obtain analogous results for orthogonal groups, M.R. Stein defined "absolute stable range" conditions on a commutative ring R. Because he was working with group schemes, Stein did not consider absolute stable range conditions for noncommutative rings. Here we do so, and take up a corresponding stability question for orthogonal groups, namely cancellation of quadratic forms. For this we use a very general definition of quadratic form, which specializes to all classical examples.Sections 1, 2 and 3 contain definitions associated with, and computations of, absolute stable rank. Definitions associated with quadratic forms are introduced in Sections 4, 5, 6 and 7; and Section 8 is devoted to Witt cancellation.

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