Some results on homogeneous locally nilpotent $R$-derivations on $R[X,Y,Z]$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916168531443712 |
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| author | Ghosh, Parnashree |
| author_facet | Ghosh, Parnashree |
| contents | Let $k$ be a field of characteristic zero and $R$ a $k$-algebra. In this paper we study homogeneous $R$-lnds $D$ on $R[X,Y,Z]$ with respect to the standard weights $(1,1,1)$. We show that when $R$ is a PID, $rank(D)$ can be at most $2$ if $°(D) \leqslant 3$. As a consequence we obtain a certain class of homogeneous lnds on $k^{[4]}$ whose kernel is $k^{[3]}$. Further when $R$ is a Dedekind domain, we give a bound for minimum number of generators of $\ker(D)$ as an $R$-algebra if $°(D) \leqslant 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_12630 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Some results on homogeneous locally nilpotent $R$-derivations on $R[X,Y,Z]$ Ghosh, Parnashree Commutative Algebra 13N15, 13F20 Let $k$ be a field of characteristic zero and $R$ a $k$-algebra. In this paper we study homogeneous $R$-lnds $D$ on $R[X,Y,Z]$ with respect to the standard weights $(1,1,1)$. We show that when $R$ is a PID, $rank(D)$ can be at most $2$ if $°(D) \leqslant 3$. As a consequence we obtain a certain class of homogeneous lnds on $k^{[4]}$ whose kernel is $k^{[3]}$. Further when $R$ is a Dedekind domain, we give a bound for minimum number of generators of $\ker(D)$ as an $R$-algebra if $°(D) \leqslant 3$. |
| title | Some results on homogeneous locally nilpotent $R$-derivations on $R[X,Y,Z]$ |
| topic | Commutative Algebra 13N15, 13F20 |
| url | https://arxiv.org/abs/2202.12630 |