Geodesic
Homology of the full linear group over a~local ring, and Milnor's $K$-theory
Izvestiya. Mathematics , Tome 34 (1990) no. 1, pp. 121-145.

Voir la notice de l'article provenant de la source Math-Net.Ru

For rings with a large number of units the authors prove a strengthened theorem on homological stabilization: the homomorphism $H_k(\operatorname{GL}_n(A))\to H_k(\operatorname{GL}(A))$ is surjective for $n\geqslant k+\operatorname{sr}A-1$ and bijective for $n\geqslant k+\operatorname{sr}A$. If $A$ is a local ring with an infinite residue field, then this result admits further refinement: the homomorphism $H_n(\operatorname{GL}_n(A))\to H_n(\operatorname{GL}(A))$ is bijective and the factor group $H_n(\operatorname{GL}(A))/H_n(\operatorname{GL}_{n-1}(A))$ is canonically isomorphic to Milnor's $n$ th $K$-group of the ring $A$. The results are applied to compute the Chow groups of algebraic varieties. Bibliography: 16 titles. 
@article{IM2_1990_34_1_a5,
     author = {Yu. P. Nesterenko and A. A. Suslin},
     title = {Homology of the full linear group over a~local ring, and {Milnor's} $K$-theory},
     journal = {Izvestiya. Mathematics },
     pages = {121--145},
     publisher = {mathdoc},
     volume = {34},
     number = {1},
     year = {1990},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1990_34_1_a5/}
}
TY  - JOUR
AU  - Yu. P. Nesterenko
AU  - A. A. Suslin
TI  - Homology of the full linear group over a~local ring, and Milnor's $K$-theory
JO  - Izvestiya. Mathematics 
PY  - 1990
SP  - 121
EP  - 145
VL  - 34
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1990_34_1_a5/
LA  - en
ID  - IM2_1990_34_1_a5
ER  - 
%0 Journal Article
%A Yu. P. Nesterenko
%A A. A. Suslin
%T Homology of the full linear group over a~local ring, and Milnor's $K$-theory
%J Izvestiya. Mathematics 
%D 1990
%P 121-145
%V 34
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1990_34_1_a5/
%G en
%F IM2_1990_34_1_a5
Yu. P. Nesterenko; A. A. Suslin. Homology of the full linear group over a~local ring, and Milnor's $K$-theory. Izvestiya. Mathematics , Tome 34 (1990) no. 1, pp. 121-145. http://geodesic.mathdoc.fr/item/IM2_1990_34_1_a5/

[1] Bass X., Algebraicheskaya $K$-teoriya, Mir, M., 1973, 591 pp. | MR | Zbl

[2] Vasershtein L. N., “Stabilnyi rang kolets i razmernost topologicheskikh prostranstv”, Funkts. analiz i ego prilozh., 5:2 (1971), 17–27 | MR | Zbl

[3] Grotendik A., O nekotorykh voprosakh gomologicheskoi algebry, Mir, M., 1961, 175 pp.

[4] Maklein S., Gomologiya, Mir, M., 1966, 543 pp.

[5] Milnor Dzh., Vvedenie v algebraicheskuyu $K$-teoriyu, Mir, M., 1974, 197 pp. | MR | Zbl

[6] Suslin A. A., “Gomologii $GL_n$, kharakteristicheskie klassy i $K$-teoriya Milnora”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 165, 1984, 188–204 | MR | Zbl

[7] Bass H., Tate J., “The Milnor ring of a global field”, Lect. Notes Math., 342, 1973, 349–446 | MR | Zbl

[8] Block S., Algebraic cycles and higher $K$-theory, Preprint, 1985

[9] Brown K. S., Cohomology of groups, Springer-Verlag, 1982, 306 pp. | MR

[10] Cartan H., Algebres d'Eilenberg–Maclain et Homotopie, Séminaire H. Cartan, Ecole Norm. Super. Paris, 1954/55

[11] Kallen van der W., “The $K_2$ of rings with many units”, Ann. Sci. Ecole Norm. Super, 10 (1977), 473–515 | MR | Zbl

[12] Kallen van der W., “Homology stability for general linear groups”, Invent. Math., 60:3 (1980), 269–295 | DOI | MR | Zbl

[13] Quillen D., “Higher $K$-theory, I”, Lect. Notes Math., 341, 1973, 89–147

[14] Soulé Ch., “Operations en $K$-theorie algébrique”, Canadian J. of Math., 37:3 (1985), 488–550 | MR | Zbl

[15] Suslin A. A., “Mennicke symbols and their applications in the $K$-theory of fields”, Lect. Notes Math., 966, 1982, 334–357 | MR

[16] Suslin A. A., “Stability in algebraic $K$-theory”, Lect. Notes Math., 966, 1982, 304–334 | MR