Analytic spread of filtrations and symbolic algebras

Abstract

In this paper we define and explore the analytic spread 𝓁() 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, we have the classical bounds ht(𝐼) ⩽ 𝓁(𝐼) ⩽ dim 𝑅. The upper bound 𝓁() ⩽ dim 𝑅 is true for filtrations , but the lower bound is not true for all filtrations. We show that for the filtration  of symbolic powers of a height two prime ideal 𝔭 in a regular local ring of dimension three (a space curve singularity), so that ht() = 2 and dim 𝑅 = 3, we have that 0 ⩽ 𝓁() ⩽ 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 for all symbolic powers of 𝔭 and if and only if 𝓁( 𝑎 ) = 3 for all truncations  𝑎 of .

Analytic Spread of Filtrations on Two-Dimensional Normal Local Rings

“…The Rees algebra of the filtration is . Analogously to the case of ideals, we define the fiber cone of the filtration to be and the analytic spread of the filtration of to be We have that for all n [12, (7)], so it is natural to define .…”

“…3.6] that so the second inequality of Theorem 1.1 always holds. However, the first inequality of Theorem 1.1, fails spectacularly, even attaining the condition that [12, Exams. 1.2, 6.1, and 6.6].…”

“…Example 7.3 is of a symbolic algebra of an intersection of height 1 prime ideals in a two-dimensional excellent normal local ring. In the case that is a Noetherian filtration ( is a finitely generated R -algebra), the lower bound always holds [12, Prop. 3.7], so that the inequality of Theorem 1.1 for ideals continues to hold for Noetherian filtrations.…”

“…The condition that a filtration has analytic spread zero has a simple ideal theoretic interpretation (see [12, Lem. 3.8]).…”

“…This example is of a symbolic filtration where are height 1 prime ideals in a two-dimensional normal excellent local ring R . In contrast, since the filtration is not Noetherian, we have (by [12, Cor. 1.9]) that for every , the analytic spread of the ideal is , the largest possible.…”