Journal of Algebraic Geometry

Author(s): Moritz Kerz
Journal: J. Algebraic Geom. 19 (2010), 173-191.
Posted: July 9, 2009
Retrieve article in: PDF

Abstract: We propose a definition of improved Milnor

-groups of local rings with finite residue fields, such that the improved Milnor

-sheaf in the Zariski topology is a universal extension of the naive Milnor

-sheaf with a certain transfer map for étale extensions of local rings. The main theorem states that the improved Milnor

-ring is generated by elements of degree one.

1.: Elbaz-Vincent, Philippe; Müller-Stach, Stefan; Milnor -theory of rings, higher Chow groups and applications. Invent. Math. 148, (2002), no. 1, 177-206. MR 1892848 (2003c:19001)
2.: Gabber, Ofer; Letter to Bruno Kahn, 1998.
3.: Gabber, Ofer; -theory of Henselian local rings and Henselian pairs. (Santa Margherita Ligure, 1989), 59-70, Contemp. Math., 126, Amer. Math. Soc., Providence, RI, 1992. MR 1156502 (93c:19005)
4.: Grothendieck, A.; Eléments de géométrie algébrique IV. Inst. Hautes Etudes Sci. Publ. Math.
5.: Hoobler, R.; The Merkuriev-Suslin theorem for any semi-local ring. -theory Preprint Archives, 731.
6.: Kahn, Bruno; Deux théorèmes de comparaison en cohomologie étale. Duke Math. J. 69 (1993), 137-165. MR 1201695 (94g:14009)
7.: Kahn, Bruno; The Quillen-Lichtenbaum conjecture at the prime 2. Preprint (1997), K-theory preprint Archives, 208.
8.: Kato, Kazuya; Milnor -theory and the Chow group of zero cycles. (Boulder, Colo., 1983), 241-253, Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 1986. MR 862638 (88c:14012)
9.: Kerz, Moritz; The Gersten conjecture for Milnor -theory. Preprint (2006), K-theory preprint Archives, 791.
10.: Kerz, Moritz; Müller-Stach, Stefan; The Milnor-Chow homomorphism revisited. -Theory 38 (2007), no. 1, 49-58. MR 2353863 (2009e:14014)
11.: Kolster, Manfred; $K\sb 2$ of noncommutative local rings. J. Algebra 95 (1985), no. 1, 173-200. MR 797662 (86k:16021)
12.: Mazza, C.; Voevodsky, V.; Weibel, Ch.; Lecture notes on motivic cohomology. Clay Mathematics Monographs, Cambridge, MA, 2006. MR 2242284 (2007e:14035)
13.: Maazen, H.; Stienstra, J.; A presentation for $K\sb{2}$ of split radical pairs. J. Pure Appl. Algebra 10 (1977/78), no. 3, 271-294. MR 0472795 (57:12485)
14.: Milnor, John; Algebraic -theory and quadratic forms. Invent. Math. 9, 1969/1970, 318-344. MR 0260844 (41:5465)
15.: Orlov, D.; Vishik, A.; Voevodsky, V.; An exact sequence for $K\sp M\sb */2$ with applications to quadratic forms. Ann. of Math. (2) 165 (2007), no. 1, 1-13. MR 2276765 (2008c:19001)
16.: Nesterenko, Yu.; Suslin, A.; Homology of the general linear group over a local ring, and Milnor's -theory. Math. USSR-Izv. 34 (1990), no. 1, 121-145. MR 992981 (90a:20092)
17.: Suslin, A.; Yarosh, V.; Milnor's $K\sb 3$ of a discrete valuation ring. Algebraic -theory, 155-170, Adv. Soviet Math., 4. MR 1124631 (92j:19003)
18.: Thomason, R. W.; Le principe de scindage et l'inexistence d'une -theorie de Milnor globale. Topology 31 (1992), no. 3, 571-588. MR 1174260 (93j:19005)
19.: van der Kallen, W.; The $K\sb{2}$ of rings with many units. Ann. Sci. Ecole Norm. Sup. (4) 10 (1977), no. 4, 473-515. MR 0506170 (58:22018)

Previous issue \| Table of contents \| Next issue Recently posted articles \| Most recent issue \| All issues