Serre dimension and Euler class groups of overrings of polynomial rings

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.