Blow-up rings and $F$-rationality
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914787994107904 |
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| author | Kotal, Nirmal Kummini, Manoj |
| author_facet | Kotal, Nirmal Kummini, Manoj |
| contents | In this paper, we prove some sufficient conditions for Cohen-Macaulay normal Rees algebras to be $F$-rational. Let $(R,\mathfrak{m})$ be a Gorenstein normal local domain of dimension $d\geq 2$ and of characteristic $p > 0$. Let $I$ be a $\mathfrak{m}$-primary ideal. Our first set of results give conditions on the test ideals $τ(I^n)$, $n \geq 1$ which would imply that the normalization of the Rees algebra $R[It]$ is $F$-rational. Another sufficient condition is that the socle of $\mathrm{H}_{\overline{G}_+}^d(\overline{G})$ (where $\overline{G}$ is the associated graded ring for the integral closure filtration) is entirely in degree $-1$, if $R$ is $F$-rational (but not necessarily Gorenstein). Then we show that if $R$ is a hypersurface of degree $2$ or is three-dimensional and $F$-rational, and $\mathrm{Proj} (R[\mathfrak{m} t ])$ is $F$-rational, then $R[\mathfrak{m} t ]$ is $F$-rational. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_12383 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Blow-up rings and $F$-rationality Kotal, Nirmal Kummini, Manoj Commutative Algebra 13A35, 13A30 In this paper, we prove some sufficient conditions for Cohen-Macaulay normal Rees algebras to be $F$-rational. Let $(R,\mathfrak{m})$ be a Gorenstein normal local domain of dimension $d\geq 2$ and of characteristic $p > 0$. Let $I$ be a $\mathfrak{m}$-primary ideal. Our first set of results give conditions on the test ideals $τ(I^n)$, $n \geq 1$ which would imply that the normalization of the Rees algebra $R[It]$ is $F$-rational. Another sufficient condition is that the socle of $\mathrm{H}_{\overline{G}_+}^d(\overline{G})$ (where $\overline{G}$ is the associated graded ring for the integral closure filtration) is entirely in degree $-1$, if $R$ is $F$-rational (but not necessarily Gorenstein). Then we show that if $R$ is a hypersurface of degree $2$ or is three-dimensional and $F$-rational, and $\mathrm{Proj} (R[\mathfrak{m} t ])$ is $F$-rational, then $R[\mathfrak{m} t ]$ is $F$-rational. |
| title | Blow-up rings and $F$-rationality |
| topic | Commutative Algebra 13A35, 13A30 |
| url | https://arxiv.org/abs/2305.12383 |