Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

On algebras which are locally $ \mathbb{A}^{1}$ in codimension-one


Authors: S. M. Bhatwadekar, Amartya K. Dutta and Nobuharu Onoda
Journal: Trans. Amer. Math. Soc. 365 (2013), 4497-4537
MSC (2010): Primary 13F20; Secondary 14R25, 13E15
Published electronically: January 9, 2013
MathSciNet review: 3066764
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ R$ be a Noetherian normal domain. Call an $ R$-algebra $ A$ ``locally $ \mathbb{A}^{1}$ in codimension-one'' if $ R_P \otimes _R A$ is a polynomial ring in one variable over $ R_P$ for every height-one prime ideal $ P$ in $ R$. We shall describe a general structure for any faithfully flat $ R$-algebra $ A$ which is locally $ \mathbb{A}^{1}$ in codimension-one and deduce results giving sufficient conditions for such an $ R$-algebra to be a locally polynomial algebra. We also give a recipe for constructing $ R$-algebras which are locally $ \mathbb{A}^{1}$ in codimension-one. When $ R$ is a normal affine spot (i.e., a normal local domain obtained by a localisation of an affine domain), we give criteria for a faithfully flat $ R$-algebra $ A$, which is locally $ \mathbb{A}^{1}$ in codimension-one, to be Krull and a further condition for $ A$ to be Noetherian. The results are used to construct intricate examples of faithfully flat $ R$-algebras locally $ \mathbb{A}^{1}$ in codimension-one which are Noetherian normal but not finitely generated.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 13F20, 14R25, 13E15

Retrieve articles in all journals with MSC (2010): 13F20, 14R25, 13E15


Additional Information

S. M. Bhatwadekar
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India
Address at time of publication: Bhaskaracharya Pratishthana, 56/14, Erandwane, Damle Path, Off Law College Road, Pune, 411 004, India
Email: smb@math.tifr.res.in, smbhatwadekar@gmail.com

Amartya K. Dutta
Affiliation: Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India
Email: amartya@isical.ac.in

Nobuharu Onoda
Affiliation: Department of Mathematics, University of Fukui, Fukui 910-8507, Japan
Email: onoda@u-fukui.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05619-X
Keywords: Codimension-one, faithfully flat, finite generation, retraction, complete local, Krull domain, divisorial ideal, symbolic power
Received by editor(s): November 12, 2010
Received by editor(s) in revised form: April 28, 2011
Published electronically: January 9, 2013
Article copyright: © Copyright 2013 American Mathematical Society