Open Access
2017 Serre dimension and Euler class groups of overrings of polynomial rings
Manoj K. Keshari, Husney Parvez Sarwar
J. Commut. Algebra 9(2): 213-242 (2017). DOI: 10.1216/JCA-2017-9-2-213


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.


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Manoj K. Keshari. Husney Parvez Sarwar. "Serre dimension and Euler class groups of overrings of polynomial rings." J. Commut. Algebra 9 (2) 213 - 242, 2017.


Published: 2017
First available in Project Euclid: 3 June 2017

zbMATH: 06726982
MathSciNet: MR3659949
Digital Object Identifier: 10.1216/JCA-2017-9-2-213

Primary: 13B25 , 13C10

Keywords: Euler class group , Serre dimension , unimodular element

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

Vol.9 • No. 2 • 2017
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