Journal of Commutative Algebra

Serre dimension and Euler class groups of overrings of polynomial rings

Manoj K. Keshari and Husney Parvez Sarwar

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Abstract

Let $R$ be a commutative Noetherian ring of dimension~$d$ and \[ B=R[X_1,\ldots ,X_m,Y_1^{\pm 1},\ldots ,Y_n^{\pm 1}] \] a Laurent polynomial ring over $R$. If $A=B[Y,f^{-1}]$ for some $f\in R[Y]$, then we prove the following results:

(i) if $f$ is a monic polynomial, then the Serre dimension of $A$ is $\leq d$. The case $n=0$ is due to Bhatwadekar, without the condition that $f$ is a monic polynomial.

(ii) The $p$th Euler class group $E^p(A)$ of $A$, defined by Bhatwadekar and Sridharan, is trivial for $p\geq \max \{d+1,\dim A -p+3\}$. The case $m=n=0$ is due to Mandal and Parker.

Article information

Source
J. Commut. Algebra, Volume 9, Number 2 (2017), 213-242.

Dates
First available in Project Euclid: 3 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.jca/1496476822

Digital Object Identifier
doi:10.1216/JCA-2017-9-2-213

Mathematical Reviews number (MathSciNet)
MR3659949

Zentralblatt MATH identifier
06726982

Subjects
Primary: 13B25: Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10] 13C10: Projective and free modules and ideals [See also 19A13]

Keywords
Euler class group Serre dimension unimodular element

Citation

Keshari, Manoj K.; Sarwar, Husney Parvez. Serre dimension and Euler class groups of overrings of polynomial rings. J. Commut. Algebra 9 (2017), no. 2, 213--242. doi:10.1216/JCA-2017-9-2-213. https://projecteuclid.org/euclid.jca/1496476822


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