On algebras which are locally 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 be a Noetherian normal domain. Call an -algebra ``locally in codimension-one'' if is a polynomial ring in one variable over for every height-one prime ideal in . We shall describe a general structure for any faithfully flat -algebra which is locally in codimension-one and deduce results giving sufficient conditions for such an -algebra to be a locally polynomial algebra. We also give a recipe for constructing -algebras which are locally in codimension-one. When 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 -algebra , which is locally in codimension-one, to be Krull and a further condition for to be Noetherian. The results are used to construct intricate examples of faithfully flat -algebras locally 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

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