Euler class group of certain overrings of a polynomial ring

Abstract

Let $A$ be a commutative Noetherian ring of dimension~$n$ and $P$ a projective $A$-module of rank~$n$ with trivial determinant. In \cite {BR1}, Bhatwadekar and Sridharan defined the $n$th Euler class group of~$A$ and studied the obstruction to the existence of unimodular element in~$P$. For $R=A[T]$ and $R=A[T,T^{-1}]$, the $n$th Euler class groups of $R$ are defined by Das and Keshari in \cite {mkd1, MK1}, under certain assumption on $A$ in the latter case. We define the $n$th Euler class group of the ring $R=A[T,1/f(T)]$, where $f(T)\in A[T]$ is a monic polynomial and the height of the Jacobson radical of $A$ is $\geq 2$. Also, we prove results similar to those in \cite {MK1}.