# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [1] Bass, H, K-theory and stable algebra, Inst. hautes études sci. publ. math., 22, 489-544, (1964) · Zbl 0248.18025 [2] Bass, H; Tate, J, The Milnor ring of a global field, (), 349-446 · Zbl 0299.12013 [3] Brown, K, Cohomology of groups, () · Zbl 0367.18012 [4] Guin, D, Stabilité de l’homologie du groupe linéaire et K-théorie algébrique, C. R. acad. sci. Paris, 304, 219-222, (1987) · Zbl 0609.18005 [5] Guin, D, Cohomologie et homologie non abéliennes des groupes, C. R. acad. sci. Paris, 301, 337-340, (1985) · Zbl 0584.18006 [6] Guin, D; Guin, D, Cohomologie et homologie non abéliennes des groupes, J. pure appl. algebra, 50, 109-137, (1988), IRMA Strasbourg, preprint · Zbl 0653.20051 [7] Guin, D, Homologie du groupe linéaire et symboles en K-théorie algébrique, Thèse, (1987), Strasbourg [8] Kolster, M, K2 of non commutative local rings, J. algebra, 95, 173-200, (1985) · Zbl 0588.16019 [9] Krusemeyer, M.I, Fundamental groups, algebraic K-theory, and a problem of abhyankar, Invent. math., 19, 15-47, (1973) · Zbl 0247.14005 [10] Loday, J-L, K-théorie algébrique et représentations de groupes, Ann. sci. ecole norm. sup., 9, 309-377, (1976) · Zbl 0362.18014 [11] Loday, J-L, Comparaison des homologies du groupe linéaire et de son algèbre de Lie, Ann. inst. Fourier (Grenoble), 37, 4, 167-190, (1987) · Zbl 0619.20025 [12] Loday, J-L; Quillen, D, Cyclic homology and the Lie algebra homology of matrices, Comment. math. helv., 59, 565-591, (1984) · Zbl 0565.17006 [13] Milnor, J, Algebraic K-theory and quadratic forms, Invent. math., 9, 318-344, (1970) · Zbl 0199.55501 [14] Milnor, J, Introduction to algebraic K-theory, () · Zbl 0237.18005 [15] Sah, C.H, Homology of classical Lie groups made discrete. I. stability theorems and Schur multipliers, Comment. math. helv., 61, 308-347, (1986) · Zbl 0607.57025 [16] Suslin, A.A, Homology of GLn, characteristic classes and Milnor K-theory, (), 357-384 [17] Suslin, A.A, Stability in algebraic K-theory, (), 304-333, Part I · Zbl 0498.18008 [18] van der Kallen, W, Homology stability of linear groups, Invent. math., 60, 269-295, (1980) · Zbl 0415.18012 [19] van der Kallen, W, The K2 of rings with many units, Ann. sci. ecole norm. sup., 10, 473-515, (1977) · Zbl 0393.18012 [20] Vaserstein, L.N; Vaserstein, L.N, Stable range of rings and dimension of topological spaces, Funktsional anal. i prilozhen., Funct. anal. appl., 5, 102-110, (1971) · Zbl 0239.16028 [21] Vaserstein, L.N, Bass’s first stable range condition, J. pure appl. algebra, 34, 319-330, (1984) · Zbl 0547.16017 [22] Vaserstein, L.N; Vaserstein, L.N, On the stabilization of the general linear group over a ring, Math. sb., Math. USSR-sb., 8, 383-400, (1969) · Zbl 0238.20057 [23] \scP. Vogel, à paraître.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.