On Stably Free Modules over Affine Algebras
Publications Mathématiques de l'IHÉS, Tome 116 (2012), pp. 223-243.

If X is a smooth affine variety of dimension d over an algebraically closed field k, and if (d−1)!∈k × then any stably trivial vector bundle of rank (d−1) over X is trivial. The hypothesis that X is smooth can be weakened to X is normal if d≥4.

DOI : 10.1007/s10240-012-0041-y
Fasel, J. 1 ; Rao, R. A. 2 ; Swan, R. G. 3

1 Mathematisches Institut der Universität München Theresienstrasse 39, 80333, München Germany
2 Tata Institute of Fundamental Research 1, Dr. Homi Bhabha Road, Navy Nagar, Mumbai, 400 005 India
3 University of Chicago Chicago, IL, 60637 USA
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Fasel, J.; Rao, R. A.; Swan, R. G. On Stably Free Modules over Affine Algebras. Publications Mathématiques de l'IHÉS, Tome 116 (2012), pp. 223-243. doi : 10.1007/s10240-012-0041-y. http://www.numdam.org/articles/10.1007/s10240-012-0041-y/

[1.] Kristinn Arason, J.; Pfister, A. Beweis des Krullschen Durchschnittsatzes für den Wittring, Invent. Math., Volume 12 (1971), pp. 173-176 | DOI | MR | Zbl

[2.] Balmer, P. An introduction to triangular Witt groups and a survey of applications, Algebraic and Arithmetic Theory of Quadratic Forms (Contemp. Math., 344), Amer. Math. Soc., Providence (2004), pp. 31-58 | DOI | MR | Zbl

[3.] Balmer, P.; Walter, C. A Gersten-Witt spectral sequence for regular schemes, Ann. Sci. Éc. Norm. Super., Volume 35 (2002) no. 1, pp. 127-152 | Numdam | MR | Zbl

[4.] Barge, J.; Lannes, J. Suites de Sturm, indice de Maslov et périodicité de Bott, Progress in Mathematics, 267, Birkhäuser, Basel, 2008 | MR | Zbl

[5.] Bass, H. K-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math., Volume 22 (1964), pp. 5-60 | DOI | Numdam | MR | Zbl

[6.] Bass, H. Algebraic K-theory, Benjamin, New York, 1968 | MR | Zbl

[7.] Bass, H. Libération des modules projectifs sur certains anneaux de polynômes, Séminaire Bourbaki, 26e année (1973/1974), Exp. No. 448 (Lecture Notes in Math., 431), Springer, Berlin (1975), pp. 228-354 | Numdam | MR | Zbl

[8.] Bhatwadekar, S. M. A cancellation theorem for projective modules over affine algebras over C1-fields, J. Pure Appl. Algebra, Volume 1–3 (2003), pp. 17-26 | DOI | MR | Zbl

[9.] Bloch, S. Torsion algebraic cycles, K 2, and Brauer groups of function fields, The Brauer Group (Sem., Les Plans-sur-Bex, 1980) (Lecture Notes in Math., 844), Springer, Berlin (1981), pp. 75-102 | MR | Zbl

[10.] Bloch, S.; Ogus, A. Gersten’s conjecture and the homology of schemes, Ann. Sci. Éc. Norm. Super., Volume 7 (1975), pp. 181-201 (1974.) | Numdam | MR | Zbl

[11.] Colliot-Thélène, J.-L.; Scheiderer, C. Zero-cycles and cohomology on real algebraic varieties, Topology, Volume 35 (1996) no. 2, pp. 533-559 | DOI | MR | Zbl

[12.] Fasel, J.; Srinivas, V. A vanishing theorem for oriented intersection multiplicities, Math. Res. Lett., Volume 15 (2008) no. 3, pp. 447-458 | MR | Zbl

[13.] Fasel, J.; Srinivas, V. Chow-Witt groups and Grothendieck-Witt groups of regular schemes, Adv. Math., Volume 221 (2009), pp. 302-329 | DOI | MR | Zbl

[14.] Fasel, J. Stably free modules over smooth affine threefolds, Duke Math. J., Volume 156 (2011), pp. 33-49 | DOI | MR | Zbl

[15.] Grayson, D. R. Universal exactness in algebraic K-theory, J. Pure Appl. Algebra, Volume 36 (1985), pp. 139-141 | DOI | MR | Zbl

[16.] Hornbostel, J. Constructions and dévissage in Hermitian K-theory, K-Theory, Volume 26 (2002), pp. 139-170 | DOI | MR | Zbl

[17.] Karoubi, M. Périodicité de la K-théorie hermitienne, Algebraic K-theory, III: Hermitian K-Theory and Geometric Applications (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) (Lecture Notes in Math., 343), Springer, Berlin (1973), pp. 301-411 | Zbl

[18.] Karoubi, M. Le théorème fondamental de la K-théorie hermitienne, Ann. Math., Volume 112 (1980), pp. 259-282 | DOI | MR | Zbl

[19.] Mohan Kumar, N. Stably free modules, Am. J. Math., Volume 107 (1985), pp. 1439-1444 | DOI | MR | Zbl

[20.] Merkur´ev, A. S.; Suslin, A. A. K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat., Volume 46 (1982), pp. 1011-1046 (1135–1136) | MR | Zbl

[21.] Milne, J. S. Étale cohomology, Princeton Mathematical Series, 33, Princeton University Press, Princeton, 1980 | Zbl

[22.] Milnor, J. Algebraic K-theory and quadratic forms, Invent. Math., Volume 9 (1969/1970), pp. 318-344 | DOI | MR | Zbl

[23.] Milnor, J. Introduction to algebraic K-theory, Annals of Mathematics Studies, 72, Princeton University Press, Princeton, 1971 | Zbl

[24.] Orlov, D.; Vishik, A.; Voevodsky, V. An exact sequence for K M /2 with applications to quadratic forms, Ann. Math., Volume 165 (2007), pp. 1-13 | DOI | MR | Zbl

[25.] Rao, R. A. The Bass-Quillen conjecture in dimension three but characteristic 2,3 via a question of A. Suslin, Invent. Math., Volume 93 (1988), pp. 609-618 | DOI | MR | Zbl

[26.] Rao, R. A. A stably elementary homotopy, Proc. Am. Math. Soc., Volume 137 (2009), pp. 3637-3645 | DOI | Zbl

[27.] Rao, R. A.; Kallen, W. Improved stability for SK 1 and WMS d of a non-singular affine algebra, Astérisque, Volume 226 (1994), pp. 411-420 K-theory (Strasbourg, 1992) | Numdam | Zbl

[28.] Roitman, M. On unimodular rows, Proc. Am. Math. Soc., Volume 95 (1985), pp. 184-188 | DOI | MR | Zbl

[29.] Schlichting, M. Hermitian K-theory of exact categories, K-Theory, Volume 5 (2010), pp. 105-165 | DOI | MR | Zbl

[30.] Schlichting, M. The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes, Invent. Math., Volume 179 (2010), pp. 349-433 | DOI | MR | Zbl

[31.] Serre, J.-P. Cohomologie Galoisienne, Lecture Notes in Mathematics, 5, Springer, Berlin, 1994 | Zbl

[32.] Suslin, A. A. Torsion in K 2 of fields, K-Theory, Volume 1 (1987), pp. 5-29 | DOI | MR | Zbl

[33.] Suslin, A. A. A cancellation theorem for projective modules over algebras, Dokl. Akad. Nauk SSSR, Volume 236 (1977), pp. 808-811 (in Russian). | MR | Zbl

[34.] A. A. Suslin, On stably free modules, Math. U.S.S.R. Sbornik, (1977), 479–491. doi:10.1070/SM1977v031n04ABEH003717 | Zbl

[35.] Suslin, A. A. Cancellation for affine varieties. (russian) modules and algebraic groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (L.O.M.I.), Volume 114 (1982), pp. 187-195 | MR | Zbl

[36.] Suslin, A. A. Torsion in K 2 of fields, K-Theory, Volume 1 (1987), pp. 5-29 | DOI | MR | Zbl

[37.] Swan, R. G. A cancellation theorem for projective modules in the metastable range, Invent. Math., Volume 27 (1974), pp. 23-43 | DOI | MR | Zbl

[38.] Kallen, W. A module structure on certain orbit sets of unimodular rows, J. Pure Appl. Algebra, Volume 57 (1989) no. 3, pp. 281-316 | DOI | MR | Zbl

[39.] Vaserstein, L. N. Operations on orbits of unimodular vectors, J. Algebra, Volume 100 (1986), pp. 456-461 | DOI | MR | Zbl

[40.] Vaserstein, L. N.; Suslin, A. A. Serre’s problem on projective modules over polynomial rings, and algebraic k-theory, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 40 (1976), pp. 993-1054 | MR | Zbl

[41.] Voevodsky, V. Motivic cohomology with Z/2-coefficients, Publ. Math. Inst. Hautes Études Sci., Volume 98 (2003), pp. 59-104 | DOI | Numdam | MR | Zbl

[42.] Vorst, T. The general linear group of polynomial rings over regular rings, Commun. Algebra, Volume 9 (1981), pp. 499-509 | DOI | MR | Zbl

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