On quadratic forms over semilocal rings
HTML articles powered by AMS MathViewer
- by Stefan Gille PDF
- Trans. Amer. Math. Soc. 371 (2019), 1063-1082 Request permission
Abstract:
Using a recent result of Panin and Pimenov we show that several results, as for instance the linkage principle, in the algebraic theory of quadratic forms over fields also hold for quadratic forms over regular semilocal domains which contain a field of characteristic not 2. As an application we prove that the Arason and Elman presentation of the powers of the fundamental ideal of the Witt ring of a field extends to semilocal rings which contain an infinite field of characteristic not 2.References
- Jón Kr. Arason, Cohomologische invarianten quadratischer Formen, J. Algebra 36 (1975), no. 3, 448–491 (French). MR 389761, DOI 10.1016/0021-8693(75)90145-3
- Jón Kr. Arason and Richard Elman, Powers of the fundamental ideal in the Witt ring, J. Algebra 239 (2001), no. 1, 150–160. MR 1827878, DOI 10.1006/jabr.2000.8688
- Jón Kristinn Arason and Albrecht Pfister, Beweis des Krullschen Durchschnittsatzes für den Wittring, Invent. Math. 12 (1971), 173–176 (German). MR 294251, DOI 10.1007/BF01404657
- Ricardo Baeza, Quadratic forms over semilocal rings, Lecture Notes in Mathematics, Vol. 655, Springer-Verlag, Berlin-New York, 1978. MR 0491773, DOI 10.1007/BFb0070341
- Vladimir Chernousov and Ivan Panin, Purity for Pfister forms and $F_4$-torsors with trivial $g_3$ invariant, J. Reine Angew. Math. 685 (2013), 99–104. MR 3181565, DOI 10.1515/crelle-2012-0018
- Jean-Louis Colliot-Thélène, Formes quadratiques sur les anneaux semi-locaux réguliers, Bull. Soc. Math. France Mém. 59 (1979), 13–31 (French). Colloque sur les Formes Quadratiques, 2 (Montpellier, 1977). MR 532002
- Richard Elman, Nikita Karpenko, and Alexander Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications, vol. 56, American Mathematical Society, Providence, RI, 2008. MR 2427530, DOI 10.1090/coll/056
- Richard Elman and T. Y. Lam, Pfister forms and $K$-theory of fields, J. Algebra 23 (1972), 181–213. MR 302739, DOI 10.1016/0021-8693(72)90054-3
- Stefan Gille, Stephen Scully, and Changlong Zhong, Milnor-Witt $K$-groups of local rings, Adv. Math. 286 (2016), 729–753. MR 3415696, DOI 10.1016/j.aim.2015.09.014
- Raymond T. Hoobler, The Merkuriev-Suslin theorem for any semi-local ring, J. Pure Appl. Algebra 207 (2006), no. 3, 537–552. MR 2265538, DOI 10.1016/j.jpaa.2005.10.020
- Moritz Kerz, The Gersten conjecture for Milnor $K$-theory, Invent. Math. 175 (2009), no. 1, 1–33. MR 2461425, DOI 10.1007/s00222-008-0144-8
- Fabien Morel and Vladimir Voevodsky, $\textbf {A}^1$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001). MR 1813224, DOI 10.1007/BF02698831
- 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 10.1070/IM1990v034n01ABEH000610
- 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 10.4007/annals.2007.165.1
- Ivan Panin, Rationally isotropic quadratic spaces are locally isotropic, Invent. Math. 176 (2009), no. 2, 397–403. MR 2495767, DOI 10.1007/s00222-008-0168-0
- Ivan Panin and Konstantin Pimenov, Rationally isotropic quadratic spaces are locally isotropic: II, Doc. Math. Extra vol.: Andrei A. Suslin sixtieth birthday (2010), 515–523. MR 2804263
- I. Panin and U. Rehmann, A variant of a theorem by Springer, Algebra i Analiz 19 (2007), no. 6, 117–125; English transl., St. Petersburg Math. J. 19 (2008), no. 6, 953–959. MR 2411641, DOI 10.1090/S1061-0022-08-01029-7
- Michel Raynaud, Anneaux locaux henséliens, Lecture Notes in Mathematics, Vol. 169, Springer-Verlag, Berlin-New York, 1970 (French). MR 0277519
- Winfried Scharlau, Quadratic and Hermitian forms, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985. MR 770063, DOI 10.1007/978-3-642-69971-9
- Marco Schlichting, Euler class groups and the homology of elementary and special linear groups, Adv. Math. 320 (2017), 1–81. MR 3709100, DOI 10.1016/j.aim.2017.08.034
- Stephen Scully, The Artin-Springer theorem for quadratic forms over semi-local rings with finite residue fields, Proc. Amer. Math. Soc. 146 (2018), no. 1, 1–13. MR 3723116, DOI 10.1090/proc/13744
- A. A. Suslin, Torsion in $K_2$ of fields, $K$-Theory 1 (1987), no. 1, 5–29. MR 899915, DOI 10.1007/BF00533985
- Vladimir Voevodsky, Motivic cohomology with $\textbf {Z}/2$-coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104. MR 2031199, DOI 10.1007/s10240-003-0010-6
Additional Information
- Stefan Gille
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
- MR Author ID: 681802
- Email: gille@ualberta.ca
- Received by editor(s): June 26, 2015
- Received by editor(s) in revised form: February 3, 2017, and April 5, 2017
- Published electronically: July 26, 2018
- Additional Notes: This work was supported by an NSERC grant.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 1063-1082
- MSC (2010): Primary 11E81; Secondary 11E88
- DOI: https://doi.org/10.1090/tran/7270
- MathSciNet review: 3885171