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On a conjecture of M. P. Murthy

Published online by Cambridge University Press:  22 January 2016

M. Boratyński
M. Boratyński*
Affiliation:
Institute of Mathematics, Śniadeckich 8, Warsaw, Poland
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In [M] Murthy asked if every subvariety V of kn (k-any field) with a trivial conormal bundle is a (scheme theoretic) complete intersection.

In [B] we were able to prove that all such subvarieties of kn are at least set theoretic complete intersections. Mohan Kumar has shown in [MK] that if one assumes moreover that n ≥ 2 dim V + 2 then V is a complete intersection. The aim of this paper is to prove the following result which extends the Mohan Kumar Theorem in the nonsingular (and connected) case and moreover sheds some light on his bound for n.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 77 , February 1980 , pp. 41 - 45
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1980

References

[B] Boratyński, M., A note on the set-theoretic complete intersection ideals, J. of Alg. 54 (1978), 1–5.CrossRefGoogle Scholar
[B1] Boratyński, M., A proof of the Eisenbud-Evans conjecture for polynomial rings over a field (unpublished).Google Scholar
[K] Kaplansky, I., Commutative rings, The University of Chicago Press 1974.CrossRefGoogle Scholar
[M] Murthy, M. P., Complete intersections, Conference on Commutative Algebra 1975, Queen’s University 196–211.Google Scholar
[M1] Murthy, M. P., Vector bundles over affine surfaces birationally equivalent to ruled surfaces, Ann. of Math. 89 (1969), 242–253.Google Scholar
[MK] Mohan Kumar, N., On two conjectures about polynomial rings, Inv. Math. 46 (1978), 225–236.CrossRefGoogle Scholar
[Q] Quillen, D., Projective modules over polynomial rings, Inv. Math. 36 (1976), 167–171.Google Scholar
[S] Suslin, A. A., On the stably free modules, Math Sbornik 102 (1977), 537–550 (in Russian).Google Scholar