Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-29T04:00:33.319Z Has data issue: false hasContentIssue false
  • English
  • Français

Cancellation problem for projective modules over affine algebras

Published online by Cambridge University Press:  14 November 2008

Manoj Kumar Keshari
Manoj Kumar Keshari
Affiliation:
Department of Mathematics, Indian Institute of Technology Mumbai, Mumbai - 400076, India, [email protected].
Get access

Abstract

Let A be an affine algebra of dimension n over an algebraically closed field k with 1/n! ∈ k. Let P be a projective A-module of rank n − 1. Then, it is an open question due to N. Mohan Kumar, whether P is cancellative. We prove the following results:

(i) If A = R[T,T−1], then P is cancellative.

(ii) If A = R[T,1/f] or A = R[T,f1/f,…,fr/f], where f(T) is a monic polynomial and f,f1,…,fr is R[T]-regular sequence, then An−1 is cancellative. Further, if k = p, then P is cancellative.

Keywords

projective modulesaffine domaincancellation problem
Type
Research Article
Information
Journal of K-Theory , Volume 3 , Issue 3 , June 2009 , pp. 561 - 581
Copyright
Copyright © ISOPP 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bass, H., K-theory and stable algebra, IHES 22 (1964), 5–60CrossRefGoogle Scholar
2.Bhatwadekar, S.M., Cancellation theorem for projective modules over a two-dimensional ring and its polynomial extension, Compositio Math. 128 (2001), 339–359CrossRefGoogle Scholar
3.Bhatwadekar, S.M., A cancellation theorem for projective modules over affine algebras over C1-fields, JPAA 183 (2003), 17–26Google Scholar
4.Bhatwadekar, S.M., Projective modules over affine algebras, Survey articleGoogle Scholar
5.Bhatwadekar, S.M. and Roy, A., Some theorems about projective modules over polynomial rings, J. Algebra 86 (1984), 150–158CrossRefGoogle Scholar
6.Bochnak, J., Coste, M., Roy, M.-F., Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]], 36 Springer-Verlag, Berlin, 1998Google Scholar
7.Keshari, M.K., A note on projective modules over real affine algebras, J. Algebra 278 (2004), 628–637CrossRefGoogle Scholar
8.Keshari, M.K., Stability results for projective modules over blowup rings, J. Algebra 294 (2005), 226–238CrossRefGoogle Scholar
9.Keshari, M.K., Euler class group of a Laurent polynomial ring: local case, J. Algebra 308 (2) (2007), 666–685CrossRefGoogle Scholar
10.Lindel, H., Unimodular elements in projective modules, J. Algebra 172 (1995), 301–319CrossRefGoogle Scholar
11.Kumar, N. Mohan, Stably free modules, Amer. J. Math. 107 (1985), 1439–1444CrossRefGoogle Scholar
12.Kumar, N. Mohan, Murthy, M.P. and Roy, A., A cancellation theorem for projective modules over finitely generated rings, in: Hijikata, H., et al. (Eds.), Algebraic geometry and commutative algebra in honor of Masayoshi Nagata, vol. 1, (1987), 281–287Google Scholar
13.Murthy, M.P., Cancellation problem for projective modules over certain affine algebras, Proceedings of the international colloquium on Algebra, Arithmetic and Geometry, Mumbai, Narosa Publishing House (2000), 493–507Google Scholar
14.Ojanguren, M. and Parimala, R., Projective modules over real affine algebras, Math. Ann. 287 (1990), 181–184CrossRefGoogle Scholar
15.Plumstead, B., The conjectures of Eisenbud and Evans, Amer. J. Math. 105 (1983), 1417–1433CrossRefGoogle Scholar
16.Quillen, D., Projective modules over polynomial rings, Invent. Math. 36 (1976), 167–171CrossRefGoogle Scholar
17.Rao, R.A., A question of H. Bass on the cancellative nature of large projective modules over polynomial rings, Amer. J. Math. 110 (1988), 641–657CrossRefGoogle Scholar
18.Serre, J.P., Sur les modules projectifs, Sem. Dubreil-Pisot 14 (1960–1961), 1–16Google Scholar
19.Serre, J.P., Sur la dimension cohomologique des groupes profinis, Topology 3 (1968), 264–277Google Scholar
20.Suslin, A.A., Cancellation over affine varieties, J.Sov. Math. 27 (1984), 2974–2980CrossRefGoogle Scholar
21.Suslin, A.A., A cancellation theorem for projective modules over affine algebras, Sov. Math. Dokl. 18 (1977), 1281–1284Google Scholar
22.Suslin, A.A., On the structure of the special linear group over polynomial rings, Math. USSR-Izv. 11 (1977), 221–238CrossRefGoogle Scholar
23.Suslin, A.A., Projective modules over a polynomial ring are free, Sov. Math. Dokl. 17 (1976), 1160–1164Google Scholar
24.Suslin, A.A. and Vaserstein, L.N., Serre's problem on projective modules over polynomial rings and algebraic K-theory, Math. USSR, Izvestija 10 (5) (1976), 937–1001Google Scholar
25.Swan, R.G., Projective modules over Laurent polynomial rings, Trans. Amer. Math. Soc. 237 (1978), 111–120CrossRefGoogle Scholar
26.Wiemers, A., Cancellation properties of projective modules over Laurent polynomial rings, J. Algebra 156 (1993), 108–124CrossRefGoogle Scholar