Serre dimension and Euler class group of overrings of polynomial rings

TL;DR

This paper investigates the Serre dimension and Euler class groups of certain overrings of polynomial rings over Noetherian rings, extending known results to Laurent polynomial rings and specific localizations.

Contribution

It proves bounds on Serre dimension and triviality of Euler class groups for overrings of Laurent polynomial rings, generalizing previous results to broader classes of rings.

Findings

Serre dimension of A is at most d when f is monic.

Euler class group E^p(A) is trivial for p above certain bounds.

Extends known results to Laurent polynomial rings and localizations.

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 Serre dimension of A is \leq d. In case n=0, this result 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 Raja Sridharan, is trivial for p\geq max \{d+1, \dim A -p+3\}. In case m=n=0, this result is due to Mandal-Parker.