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ON THE TRIVIALITY OF AN $\mathbb A^2$-FIBRATION OVER A DVR

Part of: Ring extensions and related topics Arithmetic rings and other special rings Field extensions Affine geometry

Published online by Cambridge University Press:  12 September 2025

PARNASHREE GHOSH Open the ORCID record for PARNASHREE GHOSH [Opens in a new window]  and
NEENA GUPTA Open the ORCID record for NEENA GUPTA [Opens in a new window]
PARNASHREE GHOSH
Affiliation:
School of Mathematics Tata Institute of Fundamental Research Mumbai-400005 India pmaths@math.tifr.res.in
NEENA GUPTA*
Affiliation:
Theoretical Statistics and Mathematics Unit Indian Statistical Institute Kolkata-700108 India
*
neenag@isical.ac.in
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Abstract

In this article, we show that any $\mathbb {A}^2$-fibration over a discrete valuation ring which is also an $\mathbb {A}^2$-form is necessarily a polynomial ring. Further, we show that separable $\mathbb {A}^2$-forms over principal ideal domains are trivial.

Keywords

Polynomial ring𝔸2-fibration𝔸2-formseparable extensiondiscrete valuation ring

MSC classification

Primary: 14R25: Affine fibrations
Secondary: 12F10: Separable extensions, Galois theory 13B25: Polynomials over commutative rings 13F20: Polynomial rings and ideals; rings of integer-valued polynomials 13F10: Principal ideal rings

Information

Type
Article
Information
Nagoya Mathematical Journal , Volume 260 , December 2025 , pp. 826 - 832
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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