Uniform bounds in excellent $\mathbf{F}_p$-algebras and applications to semi-continuity
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917983661588480 |
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| author | Lyu, Shiji |
| author_facet | Lyu, Shiji |
| contents | We study two important numerical invariants, Hilbert--Kunz multiplicity and $F$-signature, on the spectrum of a Noetherian $\mathbf{F}_p$-algebra $R$ that is not necessarily $F$-finite. When $R$ is excellent, we show that the limits defining the invariants are uniform. As a consequence, we show that the $F$-signature is lower semi-continuous, and the Hilbert--Kunz multiplicity is upper semi-continuous provided $R$ is locally equidimensional. Uniform convergence is achieved via a uniform version of Cohen--Gabber theorem. We prove the results under weaker conditions than excellence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13846 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform bounds in excellent $\mathbf{F}_p$-algebras and applications to semi-continuity Lyu, Shiji Commutative Algebra We study two important numerical invariants, Hilbert--Kunz multiplicity and $F$-signature, on the spectrum of a Noetherian $\mathbf{F}_p$-algebra $R$ that is not necessarily $F$-finite. When $R$ is excellent, we show that the limits defining the invariants are uniform. As a consequence, we show that the $F$-signature is lower semi-continuous, and the Hilbert--Kunz multiplicity is upper semi-continuous provided $R$ is locally equidimensional. Uniform convergence is achieved via a uniform version of Cohen--Gabber theorem. We prove the results under weaker conditions than excellence. |
| title | Uniform bounds in excellent $\mathbf{F}_p$-algebras and applications to semi-continuity |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2503.13846 |