Euler class group of certain overrings of a polynomial ring. (English) Zbl 1409.13020

Dhorajia, Alpesh M.

J. Commut. Algebra 9, No. 3, 341-365 (2017).

Author’s abstract: Let \(A\) be a commutative Noetherian ring of dimension \(n\) and \(P\) a projective \(A\)-module of rank \(n\) with trivial determinant. In [Compos. Math. 122, No. 2, 183–222 (2000; Zbl 0999.13007)], S. M. Bhatwadekar and R. 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 [M. K. Das, J. Algebra 264, No. 2, 582–612 (2003; Zbl 1106.13300); M. K. Keshari, J. Algebra 308, No. 2, 666–685 (2007; Zbl 1117.13012)], 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 [Zbl 1117.13012].

Reviewer: Marco Fontana (Roma)

MSC:

13C10	Projective and free modules and ideals in commutative rings
13B25	Polynomials over commutative rings

Keywords:

ring theory; projective module; unimodular element

Citations:

Zbl 0999.13007; Zbl 1106.13300; Zbl 1117.13012