Abstract
We give a group structure on unimodular rows of length three over a two-dimensional ring. This is done as follows: we take two unimodular rows take their Euler classes add these in the Euler class group and take as the sum of the unimodular rows the unimodular row whose Euler class is the sum of these two classes. We use the theory of 1-cocycles to prove that this gives a group structure on the set of unimodular rows of length three. This provides a different way of looking at the results of Vaserstein and the Vaserstein symbol.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
H. Bass, \(K\)-theory and stable algebra. Inst. Hautes Études Sci. Publ. Math. 22, 5–60 (1964)
R. Basu, Topics in classical algebraic \(K\)-theory. Ph.D. thesis, School of Mathematics, Tata Institute of Fundamental Research, Mumbai (2006)
R. Basu, R. Sridharan, On Forster’s conjecture and related results. Punjab Univ. Res. J. (Sci.) 57, 13–66 (2007)
S.M. Bhatwadekar, M.K. Keshari, A question of Nori: projective generation of ideals. \(K\)-Theory 28(4), 329–351 (2003)
S.M. Bhatwadekar, R. Sridharan, Projective generation of curves in polynomial extensions of an affine domain and a question of Nori. Invent. Math. 133(1), 161–192 (1998)
S.M. Bhatwadekar, R. Sridharan, The Euler class group of a Noetherian ring. Compos. Math. 122(2), 183–222 (2000)
S.M. Bhatwadekar, R. Sridharan, On Euler classes and stably free projective modules, Algebra, Arithmetic and Geometry, Part I, II (Mumbai, 2000). Tata Institute of Fundamental Research Studies in Mathematics, vol. 16 (Tata Institute of Fundamental Research, Bombay, 2002)
S.M. Bhatwadekar, H. Lindel, R.A. Rao, The Bass-Murthy question: Serre dimension of Laurent polynomial extensions. Invent. Math. 81(1), 189–203 (1985)
S.M. Bhatwadekar, M.K. Das, S. Mandal, Projective modules over smooth real affine varieties. Invent. Math. 166(1), 151–184 (2006)
M.K. Das, R. Sridharan, Good invariants for bad ideals. J. Algebra 323(12), 3216–3229 (2010)
N.S. Gopalakrishnan, Commutative algebra (1984)
S.K. Gupta, M.P. Murthy, Suslin’s Work on Linear Groups over Polynomial Rings and Serre Problem. ISI Lecture Notes, vol. 8 (Macmillan Co. of India Ltd, New Delhi, 1980)
M.K. Keshari, Euler class group of a Noetherian ring. Ph.D. thesis, School of Mathematics, Tata Institute of Fundamental Research, Mumbai (2001)
M. Krusemeyer, Skewly completable rows and a theorem of Swan and Towber. Commun. Algebra 4(7), 657–663 (1976)
N.M. Kumar, Complete intersections. J. Math. Kyoto Univ. 17(3), 533–538 (1977)
N.M. Kumar, A note on the cancellation of reflexive modules. J. Ramanujan Math. Soc. 17(2), 93–100 (2002)
N.M. Kumar, On a theorem of Seshadri, Connected at Infinity. Texts and Readings in Mathematics, vol. 25 (Hindustan Book Agency, New Delhi, 2003), pp. 91–104
T.Y. Lam, Serre’s Problem on Projective Modules. Springer Monographs in Mathematics (Springer, Berlin, 2006)
S. MacLane, Homology. Die Grundlehren der mathematischen Wissenschaften, vol. 114, 1st edn. (Springer, Berlin, 1967)
S. Mandal, Homotopy of sections of projective modules. J. Algebr. Geom. 1(4), 639–646 (1992). With an appendix by Madhav V. Nori
S. Mandal, Projective Modules and Complete Intersections. Lecture Notes in Mathematics, vol. 1672 (Springer, Berlin, 1997)
M.P. Murthy, R.G. Swan, Vector bundles over affine surfaces. Invent. Math. 36, 125–165 (1976)
D. Quillen, Projective modules over polynomial rings. Invent. Math. 36, 167–171 (1976)
J.P. Serre, Sur les modules projectifs. SÃl’minaire Dubreil. AlgÃÃbre et thÃl’orie des nombres 14, 1–16 (1960–1961)
C.S. Seshadri, Triviality of vector bundles over the affine space \(K^{2}\). Proc. Natl. Acad. Sci. USA 44, 456–458 (1958)
C.S. Seshadri, Algebraic vector bundles over the product of an affine curve and the affine line. Proc. Am. Math. Soc. 10, 670–673 (1959)
A.A. Suslin, Stably free modules. Mat. Sb. (N.S.) 102(144)(4), 537–550, 632 (1977)
R.G. Swan, Algebraic vector bundles on the \(2\)-sphere. Rocky Mt. J. Math. 23(4), 1443–1469 (1993)
R.G. Swan, J. Towber, A class of projective modules which are nearly free. J. Algebra 36(3), 427–434 (1975)
W. van der Kallen, A group structure on certain orbit sets of unimodular rows. J. Algebra 82(2), 363–397 (1983)
W. van der Kallen, A module structure on certain orbit sets of unimodular rows. J. Pure Appl. Algebra 57(3), 281–316 (1989)
L.N. Vaserstein, Stabilization of unitary and orthogonal groups over a ring with involution. Mat. Sb. (N.S.) 81 (123), 328–351 (1970)
L.N. Vaserstein, A.A. Suslin, Serre’s problem on projective modules over polynomial rings, and algebraic \(K\)-theory. Izv. Akad. Nauk SSSR Ser. Mat. 40(5), 993–1054, 1199 (1976)
C.T.C. Wall, A Geometric Introduction to Topology (Addison-Wesley Publishing Co., Reading, 1972)
Acknowledgements
The authors would like to thank Professor Ravi A. Rao for his valuable support during this work. The authors would like to thank Professor Gopala Krishna Srinivasan for giving his time most generously and helping us make this paper more readable. The third named author would like to thank Professor Gopala Krishna Srinivasan for his support and advice during difficult times. The authors would also like to thank the referee for going through the paper carefully and pointing out some mistakes. The third named author also acknowledges the financial support from C.S.I.R. which enabled him to pursue his doctoral studies.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Gupta, A., Sridharan, R., Yadav, S.K. (2020). On a Group Structure on Unimodular Rows of Length Three over a Two-Dimensional Ring. In: Ambily, A., Hazrat, R., Sury, B. (eds) Leavitt Path Algebras and Classical K-Theory. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-15-1611-5_17
Download citation
DOI: https://doi.org/10.1007/978-981-15-1611-5_17
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-1610-8
Online ISBN: 978-981-15-1611-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)