Epsilon multiplicity, multiplicity=volume formula and analytic spread of family of ideals
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914531304800256 |
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| author | Sarkar, Parangama |
| author_facet | Sarkar, Parangama |
| contents | In an analytically unramified local ring $(R,\mathfrak m)$ of dimension $d\geq 1$, for a filtration of ideals $\mathfrak {I}=\{I_m\}_{m\in\mathbb N}$ satisfying $\mathfrak A(r)$ condition and for any $\mathfrak m$-primary ideal $K$, it is shown in $[18]$ that the epsilon multiplicity of the weakly graded family of ideals $\{(I_m:K)\}_{m\in\mathbb N}$ exists as a limit and it is bounded above by the epsilon multiplicity of $\mathfrak I$, $ε(\mathfrak I)$. In this article, we first show that $ε(\mathfrak I)$ coincides with the epsilon multiplicity of $\{(I_m:K)\}_{m\in\mathbb N}$ and this leads to the following: $(a)$ an expression for $ε(\mathfrak I)$ as a limit of the epsilon multiplicities of other graded families of ideals and $(b)$ a multiplicity=volume formula for the epsilon multiplicity of an ideal $I$ in $R$. In the final part of the article, we investigate the maximality of the analytic spread of filtrations of ideals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03814 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Epsilon multiplicity, multiplicity=volume formula and analytic spread of family of ideals Sarkar, Parangama Commutative Algebra In an analytically unramified local ring $(R,\mathfrak m)$ of dimension $d\geq 1$, for a filtration of ideals $\mathfrak {I}=\{I_m\}_{m\in\mathbb N}$ satisfying $\mathfrak A(r)$ condition and for any $\mathfrak m$-primary ideal $K$, it is shown in $[18]$ that the epsilon multiplicity of the weakly graded family of ideals $\{(I_m:K)\}_{m\in\mathbb N}$ exists as a limit and it is bounded above by the epsilon multiplicity of $\mathfrak I$, $ε(\mathfrak I)$. In this article, we first show that $ε(\mathfrak I)$ coincides with the epsilon multiplicity of $\{(I_m:K)\}_{m\in\mathbb N}$ and this leads to the following: $(a)$ an expression for $ε(\mathfrak I)$ as a limit of the epsilon multiplicities of other graded families of ideals and $(b)$ a multiplicity=volume formula for the epsilon multiplicity of an ideal $I$ in $R$. In the final part of the article, we investigate the maximality of the analytic spread of filtrations of ideals. |
| title | Epsilon multiplicity, multiplicity=volume formula and analytic spread of family of ideals |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2605.03814 |