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arXiv · 2202.00163

Variants of normality and steadfastness deform

Alexander Bauman·Havi Ellers·Gary Hu·Takumi Murayama·Sandra Nair·Ying Wang
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Abstract

The cancellation problem asks whether $A[X_1,X_2,\ldots,X_n] \cong B[Y_1,Y_2,\ldots,Y_n]$ implies $A \cong B$. Hamann introduced the class of steadfast rings as the rings for which a version of the cancellation problem considered by Abhyankar, Eakin, and Heinzer holds. By work of Asanuma, Hamann, and Swan, steadfastness can be characterized in terms of $p$-seminormality, which is a variant of normality introduced by Swan. We prove that $p$-seminormality and steadfastness deform for reduced Noetherian local rings. We also prove that $p$-seminormality and steadfastness are stable under adjoining formal power series variables for reduced (not necessarily Noetherian) rings. Our methods also give new proofs of the facts that normality and weak normality deform, which are of independent interest.

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Alexander Bauman, Havi Ellers, Gary Hu, Takumi Murayama, Sandra Nair, Ying Wang. 2022-02-01. Variants of normality and steadfastness deform. https://doi.org/10.1307/mmj%2F20226231

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