Analytic spread of filtrations and symbolic algebras

In this paper we define and explore the analytic spread ℓ ( I ) $\ell (\mathcal {I})$ of a filtration in local ring. We show that, especially for divisorial symbolic filtrations, some basic properties an ideal extend to even when is non-Noetherian. also illustrate significant differences between with examples. case $I$ , have classical bounds ht ⩽ dim R $\mbox{ht}(I)\leqslant \ell (I)\leqslant \dim R$ . The upper bound {I})\leqslant true filtrations $\mathcal {I}$ but lower not all filtrations. that powers height two prime p $\mathfrak {p}$ regular ring dimension three (a space curve singularity), so = 2 $\mbox{ht}(\mathcal {I}) =2$ 3 $\dim R=3$ 0 $0\leqslant 2$ values 0, 1 can occur. cases algebra necessarily non-Noetherian if only n (\mathfrak {p}^{(n)})=3$ {I}_a)=3$ truncations {I}_a$