Journal of Commutative Algebra

A note on quasi-monic polynomials and efficient generation of ideals

Md. Ali Zinna

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $A$ be a commutative Noetherian ring, and let $I$ be an ideal of $A[T]$ containing a quasi-monic polynomial. Assuming that $I/I^2$ is generated by $n$ elements, where $n\geq \dim (A[T]/I)+2$, then, it is proven that any given set of $n$ generators of $I/I^2$ can be lifted to a set of $n$ generators of $I$. It is also shown that various types of Horrocks' type results (previously proven for monic polynomials) can be generalized to the setting of quasi-monic polynomials.

Article information

Source
J. Commut. Algebra, Volume 10, Number 3 (2018), 411-433.

Dates
First available in Project Euclid: 9 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.jca/1541754168

Digital Object Identifier
doi:10.1216/JCA-2018-10-3-411

Mathematical Reviews number (MathSciNet)
MR3874661

Zentralblatt MATH identifier
06976324

Subjects
Primary: 13C10: Projective and free modules and ideals [See also 19A13] 19A15: Efficient generation

Keywords
Efficient generation of ideals projective modules quasi-monic polynomials

Citation

Zinna, Md. Ali. A note on quasi-monic polynomials and efficient generation of ideals. J. Commut. Algebra 10 (2018), no. 3, 411--433. doi:10.1216/JCA-2018-10-3-411. https://projecteuclid.org/euclid.jca/1541754168


Export citation

References

  • S.M. Bhatwadekar and Raja Sridharan, Zero cycles and the Euler class groups of smooth real affine varieties, Invent. Math. 136 (1999), 287–322.
  • ––––, Euler class group of a Noetherian ring, Compos. Math. 122 (2000), 183–222.
  • ––––, On a question of Roitman, J. Ramanujan Math. Soc. 16 (2001), 45–61.
  • M.K. Das, The Euler class group of a polynomial algebra, J. Algebra 264 (2003), 582–612.
  • M.K. Das and Md. Ali Zinna, On invariance of the Euler class group under a subintegral base change, J. Algebra 398 (2014), 131–155.
  • T.Y. Lam, Serre conjecture, Lect. Notes Math. 635, Springer, Berlin, 1978.
  • S. Mandal, On efficient generation of ideals, Invent. Math. 75 (1984), 59–67.
  • ––––, On set-theoretic intersection in affine spaces, J. Pure Appl. Alg. 51 (1988), 267–275.
  • ––––, Projective modules and complete intersections, Lect. Notes Math. 1672, Springer, Berlin, 1997.
  • N. Mohan Kumar, On two conjectures about polynomial rings, Invent. Math. 46 (1978), 225–236.
  • D. Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167–171.
  • R.A. Rao, An elementary transformation of a special unimodular vector to its top coefficients vector, Proc. Amer. Math. Soc. 93 (1985), 21–24.
  • A.A. Suslin, Projective modules over polynomial ring are free, Soviet Math. Dokl. 17 (1976), 1160–1164.
  • X. Zhu, Idempotents and quasi-monic polynomials, Comm. Alg. 31 (2003), 4853–4870.