Milnor 𝐾-theory of local rings with finite residue fields

Kerz, Moritz

doi:10.1090/S1056-3911-09-00514-1

Milnor $K$-theory of local rings with finite residue fields

Author: Moritz Kerz
Journal: J. Algebraic Geom. 19 (2010), 173-191
DOI: https://doi.org/10.1090/S1056-3911-09-00514-1
Published electronically: July 9, 2009
MathSciNet review: 2551760
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Abstract | References | Additional Information

Abstract: We propose a definition of improved Milnor $K$-groups of local rings with finite residue fields, such that the improved Milnor $K$-sheaf in the Zariski topology is a universal extension of the naive Milnor $K$-sheaf with a certain transfer map for étale extensions of local rings. The main theorem states that the improved Milnor $K$-ring is generated by elements of degree one.

References

Philippe Elbaz-Vincent and Stefan Müller-Stach, Milnor $K$-theory of rings, higher Chow groups and applications, Invent. Math. 148 (2002), no. 1, 177–206. MR 1892848, DOI https://doi.org/10.1007/s002220100193

Gabber, Ofer; Letter to Bruno Kahn, 1998.

Ofer Gabber, $K$-theory of Henselian local rings and Henselian pairs, Algebraic $K$-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989) Contemp. Math., vol. 126, Amer. Math. Soc., Providence, RI, 1992, pp. 59–70. MR 1156502, DOI https://doi.org/10.1090/conm/126/00509

Grothendieck, A.; Eléments de géométrie algébrique IV. Inst. Hautes Etudes Sci. Publ. Math.

Hoobler, R.; The Merkuriev-Suslin theorem for any semi-local ring. $K$-theory Preprint Archives, 731.

Bruno Kahn, Deux théorèmes de comparaison en cohomologie étale: applications, Duke Math. J. 69 (1993), no. 1, 137–165 (French). MR 1201695, DOI https://doi.org/10.1215/S0012-7094-93-06907-4

Kahn, Bruno; The Quillen-Lichtenbaum conjecture at the prime 2. Preprint (1997), K-theory preprint Archives, 208.

Kazuya Kato, Milnor $K$-theory and the Chow group of zero cycles, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 241–253. MR 862638, DOI https://doi.org/10.1090/conm/055.1/862638

Kerz, Moritz; The Gersten conjecture for Milnor $K$-theory. Preprint (2006), K-theory preprint Archives, 791.

Moritz Kerz and Stefan Müller-Stach, The Milnor-Chow homomorphism revisited, $K$-Theory 38 (2007), no. 1, 49–58. MR 2353863, DOI https://doi.org/10.1007/s10977-007-9006-1

Manfred Kolster, $K_2$ of noncommutative local rings, J. Algebra 95 (1985), no. 1, 173–200. MR 797662, DOI https://doi.org/10.1016/0021-8693%2885%2990100-0

Carlo Mazza, Vladimir Voevodsky, and Charles Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006. MR 2242284

Hendrik Maazen and Jan Stienstra, A presentation for $K_{2}$ of split radical pairs, J. Pure Appl. Algebra 10 (1977/78), no. 3, 271–294. MR 472795, DOI https://doi.org/10.1016/0022-4049%2877%2990007-X

John Milnor, Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1969/70), 318–344. MR 260844, DOI https://doi.org/10.1007/BF01425486

D. Orlov, A. Vishik, and V. Voevodsky, An exact sequence for $K^M_\ast /2$ with applications to quadratic forms, Ann. of Math. (2) 165 (2007), no. 1, 1–13. MR 2276765, DOI https://doi.org/10.4007/annals.2007.165.1

Yu. P. Nesterenko and A. A. Suslin, Homology of the general linear group over a local ring, and Milnor’s $K$-theory, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 121–146 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 121–145. MR 992981, DOI https://doi.org/10.1070/IM1990v034n01ABEH000610

A. A. Suslin and V. A. Yarosh, Milnor’s $K_3$ of a discrete valuation ring, Algebraic $K$-theory, Adv. Soviet Math., vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 155–170. MR 1124631

R. W. Thomason, Le principe de scindage et l’inexistence d’une $K$-theorie de Milnor globale, Topology 31 (1992), no. 3, 571–588 (French). MR 1174260, DOI https://doi.org/10.1016/0040-9383%2892%2990052-J

Wilberd van der Kallen, The $K_{2}$ of rings with many units, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 473–515. MR 506170

Additional Information

Moritz Kerz
Affiliation: NWF I-Mathematik, Universität Regensburg, 93040 Regensburg, Germany
Email: moritz.kerz@mathematik.uni-regensburg.de

Received by editor(s): October 12, 2007
Received by editor(s) in revised form: January 30, 2008
Published electronically: July 9, 2009
Additional Notes: The author is supported by Studienstiftung des deutschen Volkes.