Variants of normality and steadfastness deform
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arXiv
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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866915259944534016 |
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| author | Bauman, Alexander Ellers, Havi Hu, Gary Murayama, Takumi Nair, Sandra Wang, Ying |
| author_facet | Bauman, Alexander Ellers, Havi Hu, Gary Murayama, Takumi Nair, Sandra Wang, Ying |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_00163 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Variants of normality and steadfastness deform Bauman, Alexander Ellers, Havi Hu, Gary Murayama, Takumi Nair, Sandra Wang, Ying Commutative Algebra Algebraic Geometry 13F45, 13B25 (Primary) 14B07, 13B22, 13F25 (Secondary) 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. |
| title | Variants of normality and steadfastness deform |
| topic | Commutative Algebra Algebraic Geometry 13F45, 13B25 (Primary) 14B07, 13B22, 13F25 (Secondary) |
| url | https://arxiv.org/abs/2202.00163 |