Mennicke symbols, 𝐾-cohomology and a Bass-Kubota theorem

Fasel, J.

doi:10.1090/S0002-9947-2014-06011-X

Mennicke symbols, $K$-cohomology and a Bass-Kubota theorem
HTML articles powered by AMS MathViewer

by J. Fasel PDF

Trans. Amer. Math. Soc. 367 (2015), 191-208 Request permission

Abstract:

If $A$ is a smooth algebra of dimension $d\geq 3$ over a perfect field $k$ of characteristic different from $2$, then we show that the universal Mennicke symbol $MS_{d+1}(A)$ is isomorphic to the $K$-cohomology group $H^d(A,K_{d+1})$. We then prove an analogue of the Bass-Kubota theorem for smooth affine surfaces over the algebraic closure of a finite field.

References

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
L. Barbieri-Viale, C. Pedrini, and C. Weibel, Roitman’s theorem for singular complex projective surfaces, Duke Math. J. 84 (1996), no. 1, 155–190. MR 1394751, DOI 10.1215/S0012-7094-96-08405-7
Jean Barge and Jean Lannes, Suites de Sturm, indice de Maslov et périodicité de Bott, Progress in Mathematics, vol. 267, Birkhäuser Verlag, Basel, 2008 (French). MR 2431916, DOI 10.1007/978-3-7643-8710-5
Jean Barge and Fabien Morel, Groupe de Chow des cycles orientés et classe d’Euler des fibrés vectoriels, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 4, 287–290 (French, with English and French summaries). MR 1753295, DOI 10.1016/S0764-4442(00)00158-0
H. Bass, J. Milnor, and J.-P. Serre, Solution of the congruence subgroup problem for $\textrm {SL}_{n}\,(n\geq 3)$ and $\textrm {Sp}_{2n}\,(n\geq 2)$, Inst. Hautes Études Sci. Publ. Math. 33 (1967), 59–137. MR 244257
Rabeya Basu and Ravi A. Rao, Injective stability for $K_1$ of classical modules, J. Algebra 323 (2010), no. 4, 867–877. MR 2578583, DOI 10.1016/j.jalgebra.2009.12.012
S. M. Bhatwadekar and Raja Sridharan, On Euler classes and stably free projective modules, Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000) Tata Inst. Fund. Res. Stud. Math., vol. 16, Tata Inst. Fund. Res., Bombay, 2002, pp. 139–158. MR 1940666
S. Bloch, Torsion algebraic cycles, $K_{2}$, and Brauer groups of function fields, The Brauer group (Sem., Les Plans-sur-Bex, 1980) Lecture Notes in Math., vol. 844, Springer, Berlin, 1981, pp. 75–102. MR 611866
Spencer Bloch and Arthur Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4) 7 (1974), 181–201 (1975). MR 412191, DOI 10.24033/asens.1266
P. M. Cohn, On the structure of the $\textrm {GL}_{2}$ of a ring, Inst. Hautes Études Sci. Publ. Math. 30 (1966), 5–53. MR 207856, DOI 10.1007/BF02684355
J.-L. Colliot-Thélène and C. Scheiderer, Zero-cycles and cohomology on real algebraic varieties, Topology 35 (1996), no. 2, 533–559 (English, with English and French summaries). MR 1380515, DOI 10.1016/0040-9383(95)00015-1
J. Fasel and V. Srinivas, Chow-Witt groups and Grothendieck-Witt groups of regular schemes, Adv. Math. 221 (2009), no. 1, 302–329. MR 2509328, DOI 10.1016/j.aim.2008.12.005
Jean Fasel, The Chow-Witt ring, Doc. Math. 12 (2007), 275–312. MR 2350291
Jean Fasel, Groupes de Chow-Witt, Mém. Soc. Math. Fr. (N.S.) 113 (2008), viii+197 (French, with English and French summaries). MR 2542148, DOI 10.24033/msmf.425
J. Fasel, Projective modules over the real algebraic sphere of dimension 3, J. Algebra 325 (2011), 18–33. MR 2745527, DOI 10.1016/j.jalgebra.2010.09.039
Jean Fasel, Some remarks on orbit sets of unimodular rows, Comment. Math. Helv. 86 (2011), no. 1, 13–39. MR 2745274, DOI 10.4171/CMH/216
J. Fasel, R. A. Rao, and R. G. Swan, On stably free modules over affine algebras, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 223–243. MR 3090257, DOI 10.1007/s10240-012-0041-y
Jens Hornbostel, Oriented Chow groups, Hermitian $K$-theory and the Gersten conjecture, Manuscripta Math. 125 (2008), no. 3, 273–284. MR 2373061, DOI 10.1007/s00229-007-0148-5
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 10.1090/conm/055.1/862638
Tomio Kubota, Ein arithmetischer Satz über eine Matrizengruppe, J. Reine Angew. Math. 222 (1966), 55–57 (German). MR 188194, DOI 10.1515/crll.1966.222.55
A. S. Merkur′ev and A. A. Suslin, $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011–1046, 1135–1136 (Russian). MR 675529
James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
John Milnor, Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1969/70), 318–344. MR 260844, DOI 10.1007/BF01425486
Fabien Morel, Sur les puissances de l’idéal fondamental de l’anneau de Witt, Comment. Math. Helv. 79 (2004), no. 4, 689–703 (French, with English and French summaries). MR 2099118, DOI 10.1007/s00014-004-0815-z
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
Ravi A. Rao and Wilberd van der Kallen, Improved stability for $SK_1$ and $WMS_d$ of a non-singular affine algebra, Astérisque 226 (1994), 11, 411–420. $K$-theory (Strasbourg, 1992). MR 1317126
Markus Rost, Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319–393. MR 1418952
Claus Scheiderer, Real and étale cohomology, Lecture Notes in Mathematics, vol. 1588, Springer-Verlag, Berlin, 1994. MR 1321819, DOI 10.1007/BFb0074269
Marco Schlichting, The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes, Invent. Math. 179 (2010), no. 2, 349–433. MR 2570120, DOI 10.1007/s00222-009-0219-1
Marco Schlichting, Hermitian $K$-theory, derived equivalences and Karoubi’s fundamental theorem, arXiv:1209.0848, 2012.
Jean-Pierre Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics, Vol. 5, Springer-Verlag, Berlin-New York, 1973. Cours au Collège de France, Paris, 1962–1963; Avec des textes inédits de J. Tate et de Jean-Louis Verdier; Quatrième édition. MR 0404227, DOI 10.1007/978-3-662-21553-1
V. Srinivas, Algebraic $K$-theory, 2nd ed., Progress in Mathematics, vol. 90, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1382659, DOI 10.1007/978-0-8176-4739-1
A. A. Suslin, Mennicke symbols and their applications in the $K$-theory of fields, Algebraic $K$-theory, Part I (Oberwolfach, 1980) Lecture Notes in Math., vol. 966, Springer, Berlin-New York, 1982, pp. 334–356. MR 689382
A. A. Suslin, Torsion in $K_2$ of fields, $K$-Theory 1 (1987), no. 1, 5–29. MR 899915, DOI 10.1007/BF00533985
A. A. Suslin, The cancellation problem for projective modules, and related questions, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 323–330 (Russian). MR 562623
Wilberd van der Kallen, A module structure on certain orbit sets of unimodular rows, J. Pure Appl. Algebra 57 (1989), no. 3, 281–316. MR 987316, DOI 10.1016/0022-4049(89)90035-2
Wilberd van der Kallen, From Mennicke symbols to Euler class groups, Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000) Tata Inst. Fund. Res. Stud. Math., vol. 16, Tata Inst. Fund. Res., Bombay, 2002, pp. 341–354. MR 1940672
L. N. Vaseršteĭn, On the stabilization of the general linear group over a ring, Math. USSR-Sb. 8 (1969), 383–400. MR 0267009, DOI 10.1070/SM1969v008n03ABEH001279
Leonid N. Vaserstein, Computation of $K_1$ via Mennicke symbols, Comm. Algebra 15 (1987), no. 3, 611–656. MR 882801, DOI 10.1080/00927878708823434
L. N. Vaseršteĭn and A. A. Suslin, Serre’s problem on projective modules over polynomial rings, and algebraic $K$-theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 5, 993–1054, 1199 (Russian). MR 0447245
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
C. Weibel, The norm residue isomorphism theorem, J. Topol. 2 (2009), no. 2, 346–372. MR 2529300, DOI 10.1112/jtopol/jtp013