Let \mathcal I=\{I_n\} be a filtration of a (Noetherian) local ring R with maximal ideal m_R. The Rees algebra of \mathcal I is R[\mathcal I]=\oplus_{n\ge 0}I_n.
We define the analytic spread of \mathcal I to be \ell(\mathcal I)=\dim R[\mathcal I]/m_RR[\mathcal I], generalizing the classical definition for I-adic filtrations \{I^n\}, where I is an ideal of R. We explore this concept for arbitrary (not necessarily Noetherian) filtrations, and for divisorial filtrations, which are the natural (but not necessarily Noetherian) extension of the filtrations \{\overline{I^n}\} of integral closures of the power of an ideal I.
Many of the classical theorems for I-adic filtrations generalize to arbitrary or divisorial filtrations, but there are some interesting differences. | 
