Equidimensional morphisms onto splinters are pure
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915930929364992 |
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| author | Murayama, Takumi |
| author_facet | Murayama, Takumi |
| contents | We prove that a Noetherian ring $R$ is a splinter if and only if for every equidimensional surjective morphism $\operatorname{Spec}(S) \to \operatorname{Spec}(R)$, the map $R \to S$ is pure. This yields a large, nontrivial class of ring maps that are automatically pure. More generally, we prove that a locally Noetherian scheme $Y$ is locally a splinter if and only if every locally equidimensional morphism $X \to Y$ is strongly pure. Special cases of our results show that equidimensional fibrations over normal $\mathbf{Q}$-schemes or regular schemes of arbitrary characteristic are strongly pure. The main ingredient is a new factorization result for locally equidimensional morphisms of schemes, which is of independent interest. Additionally, we prove a weak Boutot-type theorem for $F$-rationality, which says that $F$-rationality descends under pure ring maps that are locally equidimensional under universally catenary assumptions. This statement is false without the locally equidimensional hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15563 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equidimensional morphisms onto splinters are pure Murayama, Takumi Algebraic Geometry Commutative Algebra 14B25, 13B10 (Primary) 13A35, 14D06, 13D22, 14B05 (Secondary) We prove that a Noetherian ring $R$ is a splinter if and only if for every equidimensional surjective morphism $\operatorname{Spec}(S) \to \operatorname{Spec}(R)$, the map $R \to S$ is pure. This yields a large, nontrivial class of ring maps that are automatically pure. More generally, we prove that a locally Noetherian scheme $Y$ is locally a splinter if and only if every locally equidimensional morphism $X \to Y$ is strongly pure. Special cases of our results show that equidimensional fibrations over normal $\mathbf{Q}$-schemes or regular schemes of arbitrary characteristic are strongly pure. The main ingredient is a new factorization result for locally equidimensional morphisms of schemes, which is of independent interest. Additionally, we prove a weak Boutot-type theorem for $F$-rationality, which says that $F$-rationality descends under pure ring maps that are locally equidimensional under universally catenary assumptions. This statement is false without the locally equidimensional hypothesis. |
| title | Equidimensional morphisms onto splinters are pure |
| topic | Algebraic Geometry Commutative Algebra 14B25, 13B10 (Primary) 13A35, 14D06, 13D22, 14B05 (Secondary) |
| url | https://arxiv.org/abs/2512.15563 |