Euler class groups and a theorem of Roitman
type: Item
instance of: scholarly article
title: Euler class groups and a theorem of Roitman
zbMATH Open document ID: 1217.13004
author: Mrinal Kanti Das
author: Raja Sridharan
published in: Journal of Pure and Applied Algebra
publication date: 2011-04-29T00:00:00Z
review text: The authors define and study the \(n\)th Euler class group \(E^n(A[T])\) of a polynomial ring \(A[T]\) over a commutative regular domain \(A\) such that \(2n\geq\dim A+3\) (M. K. Das, one of the two authors of the present paper, has previously defined \(E^n(A[T])\) for a \(d\)-dimensional commutative Noetherian ring \(A\) containing \(\mathbf Q\) for the case \(d=n\geq 2\)). Among other results, the authors prove that there exists a canonical group homomorphism \(\Phi:E^n(A)\to E^n(A[T])\) that is left invertible. They show that this homomorphism is an isomorphism under some additional assumptions and ask whether this is true in general. The authors prove under certain assumptions, that if the canonical map \(\Phi:E^n(A)\to E^n(A[T])\) is surjective (so \(\Phi\) is an isomorphism), then the map \(\Phi:E^n(A_S)\to E^n(A_S[T])\) is also an isomorphism for every multiplicative subset \(S\) of \(A\); for example, this holds in the above-mentioned Das setting. The authors prove under certain assumptions that if \(I\) is an ideal of \(A[T]\) of height \(n\), \(\underline f=\{f_1,\dots,f_n\}\subseteq I\), and \(I=\underline f+(I^2T)\), then the set \[ N(I;\underline f)=\{s\in A\mid\exists g_1,\dots,g_n\in I_s \text{ such that } I_s= (g_1,\dots,g_n);\;g_i- f_i\in (I^2T)_s\} \] is an ideal, which they call the \textit{Nori ideal} associated to \(I\) and \((f_1,\dots,f_n)\). Several applications are presented, including a ``local global'' principle for the Euler class group, and a new proof of Nori's conjecture on projective generation of ideals; this conjecture was proved by S. M. Bhatwadekar and M. K. Keshari.
reviewed by: Moshe Roitman
Mathematics Subject Classification ID: 13C20
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 19A15
zbMATH DE Number: 5882426
zbMATH Keywords: Euler class group
zbMATH Keywords: local global principle
zbMATH Keywords: Nori ideals
zbMATH Keywords: Quillen's localization theorem
zbMATH Keywords: regular domains
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.08.014
OpenAlex ID: W2062841828
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DOI: 10.1016/J.JPAA.2010.08.014
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description: scientific article; zbMATH DE number 5882426