The Cohen-Macaulay property of invariant rings over ring of integers of a global field
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909106073239552 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $A$ be the ring of integers of global field $K$. Let $G \subseteq GL_2(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y]$ (fixing $A$). Let $R^G$ be the ring of invariants. In the equi-characteristic case we prove $R^G$ is Cohen-Macaulay. In mixed characteristic case we prove that if for all primes $p$ dividing $|G|$ the Sylow $p$-subgroup of $G$ has exponent $p$ then $R^G$ is Cohen-Macaulay. We prove a similar case if for all primes $p$ dividing $|G|$ the prime $p$ is un-ramified in $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08962 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Cohen-Macaulay property of invariant rings over ring of integers of a global field Puthenpurakal, Tony J. Commutative Algebra Number Theory Primary 13A50, Secondary 13H10 Let $A$ be the ring of integers of global field $K$. Let $G \subseteq GL_2(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y]$ (fixing $A$). Let $R^G$ be the ring of invariants. In the equi-characteristic case we prove $R^G$ is Cohen-Macaulay. In mixed characteristic case we prove that if for all primes $p$ dividing $|G|$ the Sylow $p$-subgroup of $G$ has exponent $p$ then $R^G$ is Cohen-Macaulay. We prove a similar case if for all primes $p$ dividing $|G|$ the prime $p$ is un-ramified in $K$. |
| title | The Cohen-Macaulay property of invariant rings over ring of integers of a global field |
| topic | Commutative Algebra Number Theory Primary 13A50, Secondary 13H10 |
| url | https://arxiv.org/abs/2402.08962 |