The Cohen-Macaulay property of invariant rings over ring of integers of a global field-II
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908304692740096 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $A$ be the ring of integers of a number field $K$. Let $G \subseteq GL_3(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y, Z]$ (fixing $A$) and let $S = R^G$ be the ring of invariants. Assume the Veronese subring $S^{<m>}$ of $S$ is standard graded. We prove that if for all primes $p$ dividing $|G|$, the Sylow $p$-subgroup of $G$ has exponent $p$ then for all $l \gg 0$ the Veronese subring $S^{<ml>}$ of $S$ is Cohen-Macaulay. We prove a similar result if for all primes $p$ dividing $|G|$, the prime $p$ is unramified in $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04746 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Cohen-Macaulay property of invariant rings over ring of integers of a global field-II Puthenpurakal, Tony J. Commutative Algebra Primary 13A50, Secondary 13H10 Let $A$ be the ring of integers of a number field $K$. Let $G \subseteq GL_3(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y, Z]$ (fixing $A$) and let $S = R^G$ be the ring of invariants. Assume the Veronese subring $S^{<m>}$ of $S$ is standard graded. We prove that if for all primes $p$ dividing $|G|$, the Sylow $p$-subgroup of $G$ has exponent $p$ then for all $l \gg 0$ the Veronese subring $S^{<ml>}$ of $S$ is Cohen-Macaulay. We prove a similar result if for all primes $p$ dividing $|G|$, the prime $p$ is unramified in $K$. |
| title | The Cohen-Macaulay property of invariant rings over ring of integers of a global field-II |
| topic | Commutative Algebra Primary 13A50, Secondary 13H10 |
| url | https://arxiv.org/abs/2504.04746 |