Non-finitely generated monoids corresponding to finitely generated homogeneous subalgebras
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916189014327296 |
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| author | Higashitani, Akihiro Tani, Koichiro |
| author_facet | Higashitani, Akihiro Tani, Koichiro |
| contents | The goal of this paper is to study the possible monoids appearing as the associated monoids of the initial algebra of a finitely generated homogeneous $\Bbbk$-subalgebra of a polynomial ring $\Bbbk[x_1,\ldots,x_n]$. Clearly, any affine monoid can be realized since the initial algebra of the affine monoid $\Bbbk$-algebra is itself. On the other hand, the initial algebra of a finitely generated homogeneous $\Bbbk$-algebra is not necessarily finitely generated. In this paper, we provide a new family of non-finitely generated monoids which can be realized as the initial algebras of finitely generated homogeneous $\Bbbk$-algebras. Moreover, we also provide an example of a non-finitely generated monoid which cannot be realized as the initial algebra of any finitely generated homogeneous $\Bbbk$-algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01590 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-finitely generated monoids corresponding to finitely generated homogeneous subalgebras Higashitani, Akihiro Tani, Koichiro Commutative Algebra Primary: 13E15, Secondary: 13P10, 20M25 The goal of this paper is to study the possible monoids appearing as the associated monoids of the initial algebra of a finitely generated homogeneous $\Bbbk$-subalgebra of a polynomial ring $\Bbbk[x_1,\ldots,x_n]$. Clearly, any affine monoid can be realized since the initial algebra of the affine monoid $\Bbbk$-algebra is itself. On the other hand, the initial algebra of a finitely generated homogeneous $\Bbbk$-algebra is not necessarily finitely generated. In this paper, we provide a new family of non-finitely generated monoids which can be realized as the initial algebras of finitely generated homogeneous $\Bbbk$-algebras. Moreover, we also provide an example of a non-finitely generated monoid which cannot be realized as the initial algebra of any finitely generated homogeneous $\Bbbk$-algebra. |
| title | Non-finitely generated monoids corresponding to finitely generated homogeneous subalgebras |
| topic | Commutative Algebra Primary: 13E15, Secondary: 13P10, 20M25 |
| url | https://arxiv.org/abs/2404.01590 |