Analytic spread of filtrations and symbolic algebras
Analytic spread of filtrations and symbolic algebras
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DOI：
10.1112/jlms.12643
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发表时间：
2021-04
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影响因子：
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通讯作者：
S. Cutkosky;Parangama Sarkar
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文献类型：
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作者：
S. Cutkosky;Parangama Sarkar
在本文中，我们定义并探讨了局部环中过滤的解析扩散 ℓ(I)$\ell (\mathcal {I})$。我们表明，特别是对于除数过滤和符号过滤，理想的分析扩展的一些基本属性扩展到过滤，即使过滤是非诺特式的。我们还通过示例说明了过滤的分析扩展与理想的分析扩展之间的一些显着差异。在理想 I$I$ 的情况下，我们有经典界限 ht(I)⩽ℓ(I)⩽dimR$\mbox{ht}(I)\leqslant \ell (I)\leqslant \dim R$ 。上界 ℓ(I)⩽dimR$\ell (\mathcal {I})\leqslant \dim R$ 对于过滤 I$\mathcal {I}$ 为真，但下界并非对于所有过滤都为真。我们证明，对于三维正则局部环（空间曲线奇点）中高度二素理想 p$\mathfrak {p}$ 的符号幂的过滤 I$\mathcal {I}$，使得 ht(I)=2$\mbox{ht}(\mathcal {I}) =2$ 和 dimR=3$\dim R=3$ ，我们有0⩽ℓ(I)⩽2$0\leqslant \ell (\mathcal {I})\leqslant 2$ 并且 0、1 和 2 的所有值都可能出现。在解析扩展 0 和 1 的情况下，符号代数必然是非诺特代数。符号代数是非诺特代数当且仅当对于 p$\mathfrak {p}$ 的所有符号幂 ℓ(p(n))=3$\ell (\mathfrak {p}^{(n)})=3$ 并且当且仅当对于所有截断 Ia$\mathcal ℓ(Ia)=3$\ell (\mathcal {I}_a)=3$ I$\mathcal {I}$ 的 {I}_a$ 。
In this paper we define and explore the analytic spread ℓ(I)$\ell (\mathcal {I})$ of a filtration in a local ring. We show that, especially for divisorial and symbolic filtrations, some basic properties of the analytic spread of an ideal extend to filtrations, even when the filtration is non‐Noetherian. We also illustrate some significant differences between the analytic spread of a filtration and the analytic spread of an ideal with examples. In the case of an ideal I$I$ , we have the classical bounds ht(I)⩽ℓ(I)⩽dimR$\mbox{ht}(I)\leqslant \ell (I)\leqslant \dim R$ . The upper bound ℓ(I)⩽dimR$\ell (\mathcal {I})\leqslant \dim R$ is true for filtrations I$\mathcal {I}$ , but the lower bound is not true for all filtrations. We show that for the filtration I$\mathcal {I}$ of symbolic powers of a height two prime ideal p$\mathfrak {p}$ in a regular local ring of dimension three (a space curve singularity), so that ht(I)=2$\mbox{ht}(\mathcal {I}) =2$ and dimR=3$\dim R=3$ , we have that 0⩽ℓ(I)⩽2$0\leqslant \ell (\mathcal {I})\leqslant 2$ and all values of 0, 1 and 2 can occur. In the cases of analytic spread 0 and 1 the symbolic algebra is necessarily non‐Noetherian. The symbolic algebra is non‐Noetherian if and only if ℓ(p(n))=3$\ell (\mathfrak {p}^{(n)})=3$ for all symbolic powers of p$\mathfrak {p}$ and if and only if ℓ(Ia)=3$\ell (\mathcal {I}_a)=3$ for all truncations Ia$\mathcal {I}_a$ of I$\mathcal {I}$ .