The Rees algebra and analytic spread of a divisorial filtration

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 1.6 that McAdam's theorem holds for $\mathbb Q$-divisorial filtrations in an equidimensional local ring which is essentially of finite type over a field. 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.