On Gorensteinness of associated graded rings of filtrations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912846789476352 |
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| author | Bhat, Meghana Dubey, Saipriya Masuti, Shreedevi K. Okuma, Tomohiro Verma, Jugal K. Watanabe, Kei-ichi Yoshida, Ken-ichi |
| author_facet | Bhat, Meghana Dubey, Saipriya Masuti, Shreedevi K. Okuma, Tomohiro Verma, Jugal K. Watanabe, Kei-ichi Yoshida, Ken-ichi |
| contents | Let $(A, \mathfrak{m})$ be a Gorenstein local ring, and $\mathcal{F} =\{F_n \}_{n\in \mathbb{Z}}$ a Hilbert filtration. In this paper, we give a criterion for Gorensteinness of the associated graded ring of $\mathcal{F}$ in terms of the Hilbert coefficients of $\mathcal{F}$ in some cases. As a consequence we recover and extend a result proved by Okuma, Watanabe and Yoshida. Further, we present ring-theoretic properties of the normal tangent cone of the maximal ideal of $A=S/(f)$ where $S=K[\![x_0,x_1,\ldots, x_m]\!]$ is a formal power series ring over an algebraically closed field $K$, and $f=x_0^a-g(x_1,\ldots,x_m)$, where $g$ is a polynomial with $g \in (x_1,\ldots,x_m)^b \setminus (x_1,\ldots,x_m)^{b+1}$, and $a, \, b, \, m$ are integers. We show that the normal tangent cone $\overline{G}(\mathfrak{m})$ is Cohen-Macaulay if $A$ is normal and $a \le b$. Moreover, we give a criterion of the Gorensteinness of $\overline{G}(\mathfrak{m})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14189 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Gorensteinness of associated graded rings of filtrations Bhat, Meghana Dubey, Saipriya Masuti, Shreedevi K. Okuma, Tomohiro Verma, Jugal K. Watanabe, Kei-ichi Yoshida, Ken-ichi Commutative Algebra 13D40 (Primary), 13H10 (Secondary) Let $(A, \mathfrak{m})$ be a Gorenstein local ring, and $\mathcal{F} =\{F_n \}_{n\in \mathbb{Z}}$ a Hilbert filtration. In this paper, we give a criterion for Gorensteinness of the associated graded ring of $\mathcal{F}$ in terms of the Hilbert coefficients of $\mathcal{F}$ in some cases. As a consequence we recover and extend a result proved by Okuma, Watanabe and Yoshida. Further, we present ring-theoretic properties of the normal tangent cone of the maximal ideal of $A=S/(f)$ where $S=K[\![x_0,x_1,\ldots, x_m]\!]$ is a formal power series ring over an algebraically closed field $K$, and $f=x_0^a-g(x_1,\ldots,x_m)$, where $g$ is a polynomial with $g \in (x_1,\ldots,x_m)^b \setminus (x_1,\ldots,x_m)^{b+1}$, and $a, \, b, \, m$ are integers. We show that the normal tangent cone $\overline{G}(\mathfrak{m})$ is Cohen-Macaulay if $A$ is normal and $a \le b$. Moreover, we give a criterion of the Gorensteinness of $\overline{G}(\mathfrak{m})$. |
| title | On Gorensteinness of associated graded rings of filtrations |
| topic | Commutative Algebra 13D40 (Primary), 13H10 (Secondary) |
| url | https://arxiv.org/abs/2404.14189 |