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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on $ p$-bases of a regular affine domain extension


Author: Tomoaki Ono
Journal: Proc. Amer. Math. Soc. 136 (2008), 3079-3087
MSC (2000): Primary 13B99; Secondary 14A10
Published electronically: April 30, 2008
MathSciNet review: 2407070
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Abstract: Let $ R^p\subseteq R'\subseteq R$ be a tower of commutative rings where $ R$ is a regular affine domain over an algebraically closed field of prime characteristic $ p$ and $ R'$ is a regular domain. Suppose $ R$ has a $ p$-basis $ \{\varphi_1,\dots,\varphi_r\}$ over $ R^p$ and $ [Q(R')\, :\, Q(R^p)]=p^l$ $ (1\leq l\leq r-1)$. For a subset $ \Gamma_{r-l}$ of $ R$ whose elements satisfy a certain condition on linear independence, let $ M_{\Gamma_{r-l}}$ be a set of maximal ideals $ \mathfrak{m}$ of $ R$ such that $ \Gamma_{r-l}$ is a $ p$-basis of $ R_{\mathfrak{m}}$ over $ R'_{\mathfrak{m}'}$ $ (\mathfrak{m}'=\mathfrak{m}\cap R')$. We shall characterize this set in a geometrical aspect.


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Additional Information

Tomoaki Ono
Affiliation: Tokyo Metropolitan College of Industrial Technology, 8-17-1, Minami-senju, Arakawa-ku, Tokyo 116-0003, Japan
Email: tono@kouku-k.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09338-6
PII: S 0002-9939(08)09338-6
Keywords: Grassmannian, K\"{a}hler differential, $p$-basis, Zariski open set
Received by editor(s): November 21, 2006
Received by editor(s) in revised form: July 27, 2007
Published electronically: April 30, 2008
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.