RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Izv. RAN. Ser. Mat.: Year: Volume: Issue: Page: Find

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

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.

Full text: PDF file (2644 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 519.4
MSC: 19D45

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} 

• http://mi.mathnet.ru/eng/izv1233
• http://mi.mathnet.ru/eng/izv/v53/i1/p121

 SHARE:

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
2. Philippe Elbaz-Vincent, “The indecomposable K3 of rings and homology of SL2”, Journal of Pure and Applied Algebra, 132:1 (1998), 27
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
4. Akhtar R., “Milnor K-Theory of Smooth Varieties”, K-Theory, 32:3 (2004), 269–291
5. B. Mirzaii, “Homology Stability for Unitary Groups II”, K-Theory, 36:3-4 (2005), 305
6. Behrooz Mirzaii, “Homology of GL n : injectivity conjecture for GL4 ”, Math Ann, 340:1 (2007), 159
7. Moritz Kerz, “The Gersten conjecture for Milnor K-theory”, Invent math, 2008
8. B. Mirzaii, “Third homology of general linear groups”, Journal of Algebra, 320:5 (2008), 1851
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
10. Behrooz Mirzaii, “Bloch–Wigner theorem over rings with many units”, Math Z, 2010
11. Behrooz Mirzaii, “A Note on Third Homology of GL2”, Comm. in Algebra, 39:5 (2011), 1595
12. Moritz Kerz, Shuji Saito, “Cohomological Hasse principle and motivic cohomology for arithmetic schemes”, Publ.math.IHES, 2011
13. N. A. Vavilov, A. V. Stepanov, “Linear groups over general rings. I. Generalities”, J. Math. Sci. (N. Y.), 188:5 (2013), 490–550
14. Chander K. Gupta, Waldemar Hołubowski, “Commutator subgroup of Vershik–Kerov group”, Linear Algebra and its Applications, 2012
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
16. S. O. Gorchinskiy, D. V. Osipov, “A higher-dimensional Contou-Carrère symbol: local theory”, Sb. Math., 206:9 (2015), 1191–1259
17. S. O. Gorchinskiy, D. V. Osipov, “Tangent space to Milnor $K$-groups of rings”, Proc. Steklov Inst. Math., 290:1 (2015), 26–34
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
19. S. O. Gorchinskiy, D. V. Osipov, “Iterated Laurent series over rings and the Contou-Carrère symbol”, Russian Math. Surveys, 75:6 (2020), 995–1066
20. Gupta R. Krishna A., “Zero-Cycles With Modulus and Relative K-Theory”, Ann. K-Theory, 5:4 (2020), 757–819
•  Number of views: This page: 861 Full text: 322 References: 55 First page: 3