On $\mathfrak{m}$-adic Continuity of $F$-Splitting Ratio
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910014911807488 |
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| author | Akter, Maria |
| author_facet | Akter, Maria |
| contents | We investigate the $\mathfrak{m}$-adic continuity of Frobenius splitting dimensions and ratios for divisor pairs $(R,Δ)$ in an $F$-finite local ring $(R,\mathfrak{m},k)$ of prime characteristic $p>0$. Our main result states that if $R$ is an $F$-finite, $\mathbb{Q}$-Gorenstein, Cohen-Macaulay local ring of prime characteristic $p>0$, the Frobenius splitting numbers $a^Δ_e(R)$ remain unchanged under a suitable small perturbation. Moreover, we establish a desirable inequality of Frobenius splitting dimensions under general perturbations. That is, $\dim (R/(\mathcal{P}(R/(f),Δ|_{f})))\leq \dim (R/(\mathcal{P}(R/(f+\varepsilon),Δ|_{(f+\varepsilon)})))$ for all $\varepsilon \in \mathfrak{m}^{N\gg0}$, providing an example that demonstrates strict improvement can occur. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_12174 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $\mathfrak{m}$-adic Continuity of $F$-Splitting Ratio Akter, Maria Commutative Algebra Primary: 13A35, Secondary: 13D45, 14B05, 13H10 We investigate the $\mathfrak{m}$-adic continuity of Frobenius splitting dimensions and ratios for divisor pairs $(R,Δ)$ in an $F$-finite local ring $(R,\mathfrak{m},k)$ of prime characteristic $p>0$. Our main result states that if $R$ is an $F$-finite, $\mathbb{Q}$-Gorenstein, Cohen-Macaulay local ring of prime characteristic $p>0$, the Frobenius splitting numbers $a^Δ_e(R)$ remain unchanged under a suitable small perturbation. Moreover, we establish a desirable inequality of Frobenius splitting dimensions under general perturbations. That is, $\dim (R/(\mathcal{P}(R/(f),Δ|_{f})))\leq \dim (R/(\mathcal{P}(R/(f+\varepsilon),Δ|_{(f+\varepsilon)})))$ for all $\varepsilon \in \mathfrak{m}^{N\gg0}$, providing an example that demonstrates strict improvement can occur. |
| title | On $\mathfrak{m}$-adic Continuity of $F$-Splitting Ratio |
| topic | Commutative Algebra Primary: 13A35, Secondary: 13D45, 14B05, 13H10 |
| url | https://arxiv.org/abs/2505.12174 |