Non-finitely generated monoids corresponding to finitely generated homogeneous subalgebras

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 k-subalgebra of a polynomial ring k[x1,…,xn]. Clearly, any affine monoid can be realized since the initial algebra of the affine monoid k-algebra is...

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Bibliographic Details
Published in:Journal of pure and applied algebra 2025-01, Vol.229 (1), p.107854, Article 107854
Main Authors: Higashitani, Akihiro, Tani, Koichiro
Format: Article
Language:English
Subjects:
Finitely generated
Monoid algebra
SAGBI basis
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Summary: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 k-subalgebra of a polynomial ring k[x1,…,xn]. Clearly, any affine monoid can be realized since the initial algebra of the affine monoid k-algebra is itself. On the other hand, the initial algebra of a finitely generated homogeneous k-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 k-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 k-algebra.
ISSN:0022-4049
DOI:10.1016/j.jpaa.2024.107854