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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
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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.

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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

Amartya K. Dutta
Affiliation: Stat-Math Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India

Nobuharu Onoda
Affiliation: Department of Mathematics, University of Fukui, Fukui 910-8507, Japan

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

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