Quasi-$F^{\infty}$-split height versus quasi-$F$-regular height for rational double points and graded rings
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| Format: | Preprint |
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2026
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| _version_ | 1866909983259492352 |
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| author | Takamatsu, Teppei Yoshikawa, Shou |
| author_facet | Takamatsu, Teppei Yoshikawa, Shou |
| contents | In this paper, we study a phenomenon concerning quasi-$F$-singularities: under suitable hypotheses, the finiteness of the quasi-$F^{\infty}$-split height ($\mathrm{ht}^{\infty}$) implies quasi-$F$-regularity, and moreover, $\mathrm{ht}^{\infty}$ coincides with the quasi-$F$-regular height ($\mathrm{ht}^{\mathrm{reg}}$). We establish this coincidence for two important classes of isolated Gorenstein singularities. First, we explicitly compute $\mathrm{ht}^{\infty}$ and $\mathrm{ht}^{\mathrm{reg}}$ for all rational double points, showing that every non-$F$-pure rational double point satisfies $\mathrm{ht}^\infty = \mathrm{ht}^{\mathrm{reg}}$. Second, for localizations of graded non-$F$-pure normal Gorenstein rings with $F$-rational punctured spectrum, we again obtain the equality $\mathrm{ht}^\infty = \mathrm{ht}^{\mathrm{reg}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03491 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quasi-$F^{\infty}$-split height versus quasi-$F$-regular height for rational double points and graded rings Takamatsu, Teppei Yoshikawa, Shou Commutative Algebra Algebraic Geometry In this paper, we study a phenomenon concerning quasi-$F$-singularities: under suitable hypotheses, the finiteness of the quasi-$F^{\infty}$-split height ($\mathrm{ht}^{\infty}$) implies quasi-$F$-regularity, and moreover, $\mathrm{ht}^{\infty}$ coincides with the quasi-$F$-regular height ($\mathrm{ht}^{\mathrm{reg}}$). We establish this coincidence for two important classes of isolated Gorenstein singularities. First, we explicitly compute $\mathrm{ht}^{\infty}$ and $\mathrm{ht}^{\mathrm{reg}}$ for all rational double points, showing that every non-$F$-pure rational double point satisfies $\mathrm{ht}^\infty = \mathrm{ht}^{\mathrm{reg}}$. Second, for localizations of graded non-$F$-pure normal Gorenstein rings with $F$-rational punctured spectrum, we again obtain the equality $\mathrm{ht}^\infty = \mathrm{ht}^{\mathrm{reg}}$. |
| title | Quasi-$F^{\infty}$-split height versus quasi-$F$-regular height for rational double points and graded rings |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2601.03491 |