Stably free modules over smooth affine threefolds. (English) Zbl 1209.13015
Let \(A\) be a Noetherian ring of dimension \(d\). A projective module \(P\) over \(A\) is called stably free if \(P\oplus F\) is free where \(F\) is a free module over \(A\). Under what conditions such a projective module is indeed free has been studied extensively. These are free if the rank of \(P\) is greater than \(d\) is a classical result by H. Bass [Publ. Math., Inst. Hautes Étud. Sci. 22, 5–60 (1964; Zbl 0248.18025)]. It is easy to see that this is optimal. If the ring \(A\) is an affine algebra over an algebraically closed field, A. A. Suslin proved that the same holds if the rank of the projective module is at least \(d\) [Dokl. Akad. Nauk SSSR 236, 808–811 (1977; Zbl 0395.13003)]. The reviewer showed that over such an algebra, in general, there exist stably free projective modules of rank \(d-2\) which are not free [Am. J. Math. 107, 1439–1444 (1985; Zbl 0594.13008)]. So, the only possibility that Suslin’s theorem is not optimal is in the case of rank \(d-1\). If \(d=1,2\), it is easy to see that such modules are free. So, the first non-trivial case occurs when \(d=3\). In this paper, the author proves that if \(A\) is a smooth affine algebra of dimension three over an algebraically closed field of characteristic different from \(2,3\), then stably free projective modules of rank two are free.
The reviewer believes that the author, possibly with others, has settled the case in all dimensions, though this may yet be unpublished.
The reviewer believes that the author, possibly with others, has settled the case in all dimensions, though this may yet be unpublished.
Reviewer: N. Mohan Kumar (St. Louis)
MSC:
| 13C10 | Projective and free modules and ideals in commutative rings |
| 14J30 | \(3\)-folds |
| 19A13 | Stability for projective modules |
| 14C25 | Algebraic cycles |
| 14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 19G38 | Hermitian \(K\)-theory, relations with \(K\)-theory of rings |
References:
| [1] | J. K. Arason and A. Pfister, Beweis des Krullschen Durchschnittsatzes für den Wittring , Invent. Math. 12 (1971), 173-176. · Zbl 0212.37302 · doi:10.1007/BF01404657 |
| [2] | P. Balmer and C. Walter, A Gersten-Witt spectral sequence for regular schemes , Ann. Sci. École Norm. Sup. (4) 35 (2002), 127-152. · Zbl 1012.19003 · doi:10.1016/S0012-9593(01)01084-9 |
| [3] | J. Barge and F. Morel, Groupe de Chow des cycles orientés et classe d’Euler des fibrés vectoriels , C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), 287-290. · Zbl 1017.14001 · doi:10.1016/S0764-4442(00)00158-0 |
| [4] | H. Bass, K-theory and stable algebra , Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5-60. · Zbl 0248.18025 · doi:10.1007/BF02684689 |
| [5] | -, “Libération des modules projectifs sur certains anneaux de polynômes” in Séminaire Bourbaki, 26e année (1973/1974), no. 448, Lecture Notes in Math. 431 , Springer, Berlin, 1975. · Zbl 0304.13012 |
| [6] | S. M. Bhatwadekar, A cancellation theorem for projective modules over affine algebras over \(C_ 1\) -fields, J. Pure Appl. Algebra 183 (2003), 17-26. · Zbl 1049.13004 · doi:10.1016/S0022-4049(03)00038-0 |
| [7] | S. Bloch and A. Ogus, Gersten’s conjecture and the homology of schemes , Ann. Sci. École Norm. Sup. (4) 7 (1974), 181-201. · Zbl 0307.14008 |
| [8] | J.-L. Colliot-Thélène and C. Scheiderer, Zero-cycles and cohomology on real algebraic varieties , Topology 35 (1996), 533-559. · Zbl 0882.14016 · doi:10.1016/0040-9383(95)00015-1 |
| [9] | J. Fasel, The Chow-Witt ring , Doc. Math. 12 (2007), 275-312. · Zbl 1169.14302 |
| [10] | -, Groupes de Chow-Witt , Mém. Soc. Math. Fr. (N.S.) 113 , Soc. Math. France, Montrouge, 2008. |
| [11] | J. Fasel and V. Srinivas, A vanishing theorem for oriented intersection multiplicities , Math. Res. Lett. 15 (2008), 447-458. · Zbl 1204.13016 · doi:10.4310/MRL.2008.v15.n3.a5 |
| [12] | -, Chow-Witt groups and Grothendieck-Witt groups of regular schemes , Adv. Math. 221 (2009), 302-329. · Zbl 1167.13006 · doi:10.1016/j.aim.2008.12.005 |
| [13] | M. Karoubi, Le théorème fondamental de la \(K\) -théorie hermitienne, Ann. of Math. (2) 112 (1980), 259-282. JSTOR: · Zbl 0483.18008 · doi:10.2307/1971147 |
| [14] | M. K. Keshari, Cancellation problem for projective modules over affine algebras , J. K-Theory 3 (2009), 561-581. · Zbl 1180.13012 · doi:10.1017/is008007024jkt057 |
| [15] | J. S. Milne, Étale Cohomology , Princeton Math. Ser. 33 , Princeton Univ. Press, Princeton, 1980. · Zbl 0433.14012 |
| [16] | J. Milnor, Introduction to Algebraic \(K\) -Theory, Ann. of Math. Stud. 72 , Princeton Univ. Press, Princeton, 1971. · Zbl 0237.18005 |
| [17] | N. Mohan Kumar, Some theorems on generation of ideals in affine algebras , Comment. Math. Helv. 59 (1984), 243-252. · Zbl 0559.13002 · doi:10.1007/BF02566348 |
| [18] | -, Stably free modules , Amer. J. Math. 107 (1985), 1439-1444. JSTOR: · Zbl 0594.13008 · doi:10.2307/2374411 |
| [19] | M. P. Murthy, “Cancellation problem for projective modules over certain affine algebras” in Algebra, Arithmetic and Geometry, Part I, II (Mumbai, 2000), Tata Inst. Fund. Res. Stud. Math. 16 , Tata Inst. Fund. Res., Bombay, 2002. · Zbl 1052.13005 |
| [20] | 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), 1-13. · Zbl 1124.14017 · doi:10.4007/annals.2007.165.1 |
| [21] | R. A. Rao, The Bass-Quillen conjecture in dimension three but characteristic \(\not=2,3\) via a question of A. Suslin, Invent. Math. 93 (1988), 609-618. · Zbl 0652.13009 · doi:10.1007/BF01410201 |
| [22] | -, A stably elementary homotopy , Proc. Amer. Math. Soc. 137 (2009), 3637-3645. · Zbl 1181.13008 · doi:10.1090/S0002-9939-09-09949-3 |
| [23] | R. A. Rao and W. Van Der Kallen, “Improved stability for \(SK_ 1\) and \(WMS_ d\) of a non-singular affine algebra” in \(K\)- Theory (Strasbourg, Germany, 1992) , Astérisque 226 , Soc. Math. France, Montrouge, 1994, 411-420. · Zbl 0832.19002 |
| [24] | M. Rost, Chow groups with coefficients , Doc. Math. 1 (1996), 319-393. · Zbl 0864.14002 |
| [25] | M. Schlichting, Hermitian \(K\) -theory of exact categories, J. K-Theory 5 (2010), 105-165. · Zbl 1328.19009 |
| [26] | -, Hermitian k-theory, derived equivalences and Karoubi’s fundamental theorem , preprint, 2010. |
| [27] | J.-P. Serre, Cohomologie Galoisienne , 5th ed, Lecture Notes in Math. 5 , Springer, Berlin, 1994. |
| [28] | A. A. Suslin, A cancellation theorem for projective modules over algebras , Dokl. Akad. Nauk SSSR. 236 (1977), 808-811. · Zbl 0395.13003 |
| [29] | -, “Cancellation for affine varieties” (in Russian) in Modules and Algebraic Groups , Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 114 , “Nauka” Leningrad Otdel., Leningrad, 1982, 187-195. |
| [30] | -, Cancellation over affine varieties , J. Math. Sci. 27 (1984), 2974-2980. · Zbl 0569.14006 · doi:10.1007/BF01410752 |
| [31] | R. G. Swan and J. Towber, A class of projective modules which are nearly free , J. Algebra 36 (1975), 427-434. · Zbl 0322.13009 · doi:10.1016/0021-8693(75)90143-X |
| [32] | W. Van Der Kallen, A group structure on certain orbit sets of unimodular rows , J. Algebra 82 (1983), 363-397. · Zbl 0518.20035 · doi:10.1016/0021-8693(83)90158-8 |
| [33] | -, A module structure on certain orbit sets of unimodular rows , J. Pure Appl. Algebra 57 (1989), 281-316. · Zbl 0665.18011 · doi:10.1016/0022-4049(89)90035-2 |
| [34] | 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), 993-1054. · Zbl 0338.13015 |
| [35] | V. Voevodsky, Motivic cohomology with \({\mathbf Z}/2\) -coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59-104. · Zbl 1057.14028 · doi:10.1007/s10240-003-0010-6 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
