The Briançon-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911264085639168 |
|---|---|
| author | Ma, Linquan McDonald, Peter M. G., Rebecca R. Schwede, Karl |
| author_facet | Ma, Linquan McDonald, Peter M. G., Rebecca R. Schwede, Karl |
| contents | Suppose $J = (f_1, \dots, f_n)$ is an $n$-generated ideal in any ring $R$. We prove a general Briançon-Skoda-type containment relating the integral closure $\overline{J^{n+k-1}}$ with ordinary powers $J^k$. We prove that our result implies the full Briançon-Skoda containment $\overline{J^{n+k-1}} \subseteq J^k$ for pseudo-rational singularities (for instance regular rings), and even for the weaker condition of birational derived splinters. Our methods also yield the containment $\overline{J^{n+k}} \subseteq J^k$ for Du Bois singularities and even for a characteristic-free generalization. Our \myBrianconSkoda-type theorem also implies well-known closure-based Briançon-Skoda results $\overline{J^{n+k-1}} \subseteq (J^k)^{\mathrm{cl}}$ where, for instance, $\mathrm{cl}$ is tight or plus closure in characteristic $p > 0$, or $\mathrm{ep}$ closure or extension and contraction from $\widehat{R^+}$ in mixed characteristic. Our proof relies on a study of the tensor product of the derived image of the structure sheaf of a partially normalized blowup of $J$ with the Buchsbaum-Eisenbud complex (equivalently the Eagon-Northcott complex) associated to $(f_1,\dots,f_n)^k$.
As an application of our results and methods above, we prove the uniform Artin-Rees theorem and the uniform Briançon-Skoda theorem for excellent, respectively excellent reduced, rings of finite dimension, answering conjectures of Huneke. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11540 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Briançon-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings Ma, Linquan McDonald, Peter M. G., Rebecca R. Schwede, Karl Commutative Algebra Algebraic Geometry 13B22, 14B05, 13A35, 13D02, 14E15 Suppose $J = (f_1, \dots, f_n)$ is an $n$-generated ideal in any ring $R$. We prove a general Briançon-Skoda-type containment relating the integral closure $\overline{J^{n+k-1}}$ with ordinary powers $J^k$. We prove that our result implies the full Briançon-Skoda containment $\overline{J^{n+k-1}} \subseteq J^k$ for pseudo-rational singularities (for instance regular rings), and even for the weaker condition of birational derived splinters. Our methods also yield the containment $\overline{J^{n+k}} \subseteq J^k$ for Du Bois singularities and even for a characteristic-free generalization. Our \myBrianconSkoda-type theorem also implies well-known closure-based Briançon-Skoda results $\overline{J^{n+k-1}} \subseteq (J^k)^{\mathrm{cl}}$ where, for instance, $\mathrm{cl}$ is tight or plus closure in characteristic $p > 0$, or $\mathrm{ep}$ closure or extension and contraction from $\widehat{R^+}$ in mixed characteristic. Our proof relies on a study of the tensor product of the derived image of the structure sheaf of a partially normalized blowup of $J$ with the Buchsbaum-Eisenbud complex (equivalently the Eagon-Northcott complex) associated to $(f_1,\dots,f_n)^k$. As an application of our results and methods above, we prove the uniform Artin-Rees theorem and the uniform Briançon-Skoda theorem for excellent, respectively excellent reduced, rings of finite dimension, answering conjectures of Huneke. |
| title | The Briançon-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings |
| topic | Commutative Algebra Algebraic Geometry 13B22, 14B05, 13A35, 13D02, 14E15 |
| url | https://arxiv.org/abs/2510.11540 |