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Homologie du groupe linéaire et K-théorie de Milnor des anneaux. (Homology of the linear group and Milnor’s K-theory of rings). (French) Zbl 0669.20037
Let A be an associative ring with 1. \(GL_ nA\) the group of invertible n by n matrices over A. The general theorem of Suslin says that the canonical homomorphism \(H_ k(GL_ nA)\to H_ k(GL_{n+1}A)\) is surjective for \(n\geq \max (2k,k+sr(A)-1)\) and bijective for \(n\geq \max (2k+1,k+sr(A))\), where sr(A) is the stable rank of A.
In the present paper it is shown that the above homomorphism is an isomorphism for \(n\geq k\) in the case when \(sr(A)=1\) and A has “sufficiently many” units (the conditions H1 and H2 of the paper). This allows us to identify the cokernel of this homomorphism with \(n=k-1\) with the Milnor group \(K^ M_ n(A)\). A. A. Suslin and Yu. P. Nesterenko obtained recently [Izv. Akad. Nauk SSSR, Ser. Mat. 53, No.1, 121-146 (1989)] these results for commutative local rings A with infinite residue fields. They also showed that if a commutative A satisfies H2, then the above bounds for stability can be replaced by \(n\geq k+sr(A)-1\) and \(n\geq k+sr(A)\) respectively.

MSC:
20G10 Cohomology theory for linear algebraic groups
20G35 Linear algebraic groups over adèles and other rings and schemes
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
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