The Rees algebra and analytic spread of a divisorial filtration

TL;DR

This paper studies the properties of Rees algebras and analytic spread of divisorial filtrations, extending classical theorems to broader contexts and analyzing asymptotic behaviors of related algebraic functions.

Contribution

It generalizes McAdam's theorem for $Q$-divisorial filtrations in certain local rings and explores the asymptotic properties of length functions and symbolic filtrations.

Findings

McAdam's theorem holds for $Q$-divisorial filtrations in equidimensional local rings of dimension ≤ 3.

The limsup of the first difference of length functions is finite but may not have a limit.

An example of a symbolic filtration with infinitely many Rees valuations is provided.

Abstract

In this paper we investigate some properties of Rees algebras of divisorial filtrations and their analytic spread. A classical theorem of McAdam shows that the analytic spread of an ideal $I$ in a formally equidimensional local ring is equal to the dimension of the ring if and only if the maximal ideal is an associated prime of $R/\overline{I^n}$ for some $n$. We show in Theorem 1.5 that McAdam's theorem holds for $\mathbb Q$-divisorial filtrations in an equidimensional local ring which is essentially of finite type over an excellent local ring of dimension less than or equal to 3. This generalizes an earlier result for $\mathbb Q$-divisorial filtrations in an equicharacteristic zero excellent local domain by the author. This theorem does not hold for more general filtrations. We consider the question of the asymptotic behavior of the function $n\mapsto \lambda_R(R/I_n)$ for a $\mathbb Q$-divisorial filtration $\mathcal I=\{I_n\}$ of $m_R$-primary ideals on a $d$-dimensional normal excellent local ring. It is known from earlier work of the author that the multiplicity $$ e(\mathcal I)=d! \lim_{n\rightarrow\infty}\frac{\lambda_R(R/I_n)}{n^d} $$ can be irrational. We show in Lemma 4.1 that the limsup of the first difference function $$ \limsup_{n\rightarrow\infty}\frac{\lambda_R(I_n/I_{n+1})}{n^{d-1}} $$ is always finite for a $\mathbb Q$-divisorial filtration. We then give an example in Section 4 showing that this limsup may not exist as a limit. In the final section, we give an example of a symbolic filtration $\{P^{(n)}\}$ of a prime ideal $P$ in a normal two dimensional excellent local ring which has the property that the set of Rees valuations of all the symbolic powers $P^{(n)}$ of $P$ is infinite.