Flat morphisms with regular fibers do not preserve $F$-rationality
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929368510496768 |
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| author | Quinlan-Gallego, Eamon Simpson, Austyn Singh, Anurag K. |
| author_facet | Quinlan-Gallego, Eamon Simpson, Austyn Singh, Anurag K. |
| contents | For each positive prime integer $p$ we construct a standard graded $F$-rational ring $R$, over a field $K$ of characteristic $p$, such that $R\otimes_K\overline{K}$ is not $F$-rational. By localizing we obtain a flat local homomorphism $(R, \mathfrak{m}) \to (S, \mathfrak{n})$ such that $R$ is $F$-rational, $S/\mathfrak{m} S$ is regular (in fact, a field), but $S$ is not $F$-rational. In the process we also obtain standard graded $F$-rational rings $R$ for which $R\otimes_K R$ is not $F$-rational. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_03785 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Flat morphisms with regular fibers do not preserve $F$-rationality Quinlan-Gallego, Eamon Simpson, Austyn Singh, Anurag K. Commutative Algebra For each positive prime integer $p$ we construct a standard graded $F$-rational ring $R$, over a field $K$ of characteristic $p$, such that $R\otimes_K\overline{K}$ is not $F$-rational. By localizing we obtain a flat local homomorphism $(R, \mathfrak{m}) \to (S, \mathfrak{n})$ such that $R$ is $F$-rational, $S/\mathfrak{m} S$ is regular (in fact, a field), but $S$ is not $F$-rational. In the process we also obtain standard graded $F$-rational rings $R$ for which $R\otimes_K R$ is not $F$-rational. |
| title | Flat morphisms with regular fibers do not preserve $F$-rationality |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2307.03785 |