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Izv. Akad. Nauk SSSR Ser. Mat., 1989, Volume 53, Issue 1, Pages 121–146 (Mi izv1233)  

This article is cited in 18 scientific papers (total in 18 papers)

Homology of the full linear group over a local ring, and Milnor's $K$-theory

Yu. P. Nesterenko, A. A. Suslin


Abstract: 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.

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English version:
Mathematics of the USSR-Izvestiya, 1990, 34:1, 121–145

Bibliographic databases:

UDC: 519.4
MSC: 19D45
Received: 02.04.1987

Citation: Yu. P. Nesterenko, A. A. Suslin, “Homology of the full linear group over a local ring, and Milnor's $K$-theory”, Izv. Akad. Nauk SSSR Ser. Mat., 53:1 (1989), 121–146; Math. USSR-Izv., 34:1 (1990), 121–145

Citation in format AMSBIB
\Bibitem{NesSus89}
\by Yu.~P.~Nesterenko, A.~A.~Suslin
\paper Homology of the full linear group over a~local ring, and Milnor's $K$-theory
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1989
\vol 53
\issue 1
\pages 121--146
\mathnet{http://mi.mathnet.ru/izv1233}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=992981}
\zmath{https://zbmath.org/?q=an:0684.18001|0668.18011}
\transl
\jour Math. USSR-Izv.
\yr 1990
\vol 34
\issue 1
\pages 121--145
\crossref{https://doi.org/10.1070/IM1990v034n01ABEH000610}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Burt Totaro, “MilnorK-theory is the simplest part of algebraicK-theory”, K-Theory, 6:2 (1992), 177  crossref  mathscinet  zmath
    2. Philippe Elbaz-Vincent, “The indecomposable K3 of rings and homology of SL2”, Journal of Pure and Applied Algebra, 132:1 (1998), 27  crossref
    3. Jean-Louis Cathelineau, “Homology of tangent groups considered as discrete groups and scissors congruence”, Journal of Pure and Applied Algebra, 132:1 (1998), 9  crossref
    4. Akhtar R., “Milnor K-Theory of Smooth Varieties”, K-Theory, 32:3 (2004), 269–291  crossref  mathscinet  zmath  isi
    5. B. Mirzaii, “Homology Stability for Unitary Groups II”, K-Theory, 36:3-4 (2005), 305  crossref  mathscinet  zmath  isi
    6. Behrooz Mirzaii, “Homology of GL n : injectivity conjecture for GL4 ”, Math Ann, 340:1 (2007), 159  crossref  mathscinet  zmath  isi
    7. Moritz Kerz, “The Gersten conjecture for Milnor K-theory”, Invent math, 2008  crossref  mathscinet  isi
    8. B. Mirzaii, “Third homology of general linear groups”, Journal of Algebra, 320:5 (2008), 1851  crossref
    9. Gorchinskiy S., “Notes on the Biextension of Chow Groups”, Motives and Algebraic Cycles: a Celebration in Honour of Spencer J. Bloch, Fields Institute Communications, 56, 2009, 111–148  isi
    10. Behrooz Mirzaii, “Bloch–Wigner theorem over rings with many units”, Math Z, 2010  crossref
    11. Behrooz Mirzaii, “A Note on Third Homology of GL2”, Comm. in Algebra, 39:5 (2011), 1595  crossref
    12. Moritz Kerz, Shuji Saito, “Cohomological Hasse principle and motivic cohomology for arithmetic schemes”, Publ.math.IHES, 2011  crossref
    13. N. A. Vavilov, A. V. Stepanov, “Linear groups over general rings. I. Generalities”, J. Math. Sci. (N. Y.), 188:5 (2013), 490–550  mathnet  crossref  mathscinet
    14. Chander K. Gupta, Waldemar Hołubowski, “Commutator subgroup of Vershik–Kerov group”, Linear Algebra and its Applications, 2012  crossref
    15. Behrooz Mirzaii, F.Y.. Mokari, “A Bloch–Wigner theorem over rings with many units II”, Journal of Pure and Applied Algebra, 219:11 (2015), 5078  crossref
    16. S. O. Gorchinskiy, D. V. Osipov, “A higher-dimensional Contou-Carrère symbol: local theory”, Sb. Math., 206:9 (2015), 1191–1259  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    17. S. O. Gorchinskiy, D. V. Osipov, “Tangent space to Milnor $K$-groups of rings”, Proc. Steklov Inst. Math., 290:1 (2015), 26–34  mathnet  crossref  crossref  isi  elib  elib
    18. S. O. Gorchinskiy, D. N. Tyurin, “Relative Milnor $K$-groups and differential forms of split nilpotent extensions”, Izv. Math., 82:5 (2018), 880–913  mathnet  crossref  crossref  adsnasa  isi  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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