Test Modules of Extended Rees Algebras
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909765480742912 |
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| author | Ajit, Rahul Simper, Hunter |
| author_facet | Ajit, Rahul Simper, Hunter |
| contents | Given a reduced, local ring $R$ and an ideal $\mathfrak{a}$ of positive height, we give a decomposition of the test module, $τ(ω_T, t^{-λ})$, of the extended Rees algebra, $T =R[\mathfrak{a} t, t^{-1}]$. In particular, the degree zero component of this test module is $τ(R, \mathfrak{a}^λ)$, thereby reducing the computation of test modules for non-principal ideals to the much easier case of principal ideals. Additionally, we apply our decomposition to generalize results on the $F$-rationality of Rees and extended Rees algebras [HWY02a, Conjecture 4.1],[KK21] as well as give simplified proofs of discreteness and rationality of F-jumping numbers, among other applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01693 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Test Modules of Extended Rees Algebras Ajit, Rahul Simper, Hunter Commutative Algebra Algebraic Geometry 14F18, 14B05, 13A35 Given a reduced, local ring $R$ and an ideal $\mathfrak{a}$ of positive height, we give a decomposition of the test module, $τ(ω_T, t^{-λ})$, of the extended Rees algebra, $T =R[\mathfrak{a} t, t^{-1}]$. In particular, the degree zero component of this test module is $τ(R, \mathfrak{a}^λ)$, thereby reducing the computation of test modules for non-principal ideals to the much easier case of principal ideals. Additionally, we apply our decomposition to generalize results on the $F$-rationality of Rees and extended Rees algebras [HWY02a, Conjecture 4.1],[KK21] as well as give simplified proofs of discreteness and rationality of F-jumping numbers, among other applications. |
| title | Test Modules of Extended Rees Algebras |
| topic | Commutative Algebra Algebraic Geometry 14F18, 14B05, 13A35 |
| url | https://arxiv.org/abs/2509.01693 |