The Poincaré series of a simple complete ideal of a two-dimensional regular local ring
Let \(R\) be a two-dimensional regular local ring with maximal ideal \(m\). By a result of Zariski, every non-zero \(m\)-primary complete ideal of \(R\) can be written in a unique way as a product of simple complete ideals of \(R\). On the other hand, there exists a one-to-one correspondence between simple complete ideals of \(R\) and divisorial valuations of the fraction field \(K\) of \(R\) centered at \(R\). Let's assume that a simple ideal \(p\) corresponds to a valuation \(\nu\) by the above correspondence, the ideals of \(R\), \(P_n = \{f \in R | \nu(f) \geq n \}\), \( n \in {\mathbb N}\), are named \(\nu\)-ideals and they define an interesting graded \(k\)-algebra, where \(k:=R/m\): \(gr_p R := \oplus_{n \in {\mathbb N}} P_n / P_{n+1}\) and the corresponding Poincaré series \(P_p (t) := \sum_{n \in {\mathbb N}} \dim_k ( P_n / P_{n+1}) t^n\). These objects has been also defined por finitely many simple ideals. When \(k\) is algebraically closed, the series has been computed by the reviewer [Bull. Belg. Math. Soc. - Simon Stevin 2, No. 1, 65--74 (1995; Zbl 0828.13008)] for a unique ideal \(p\), and by \textit{F. Delgado, A. Núñez} and the reviewer [Adv. Math. 219, No. 5, 1632--1655 (2008; Zbl 1154.14001)] for finitely many ones. The paper under review extends the computation of the Poincaré series of a unique simple ideal to the case when the field \(k\) is infinite. The authors of this well-written paper introduce also a motivic version of the mentioned Poincaré series and compare among them the Poincaré series of a simple complete ideal, that of the value semigroup of the attached to \(p\) divisorial valuation and the motivic one.
plane valuation, motivic Poincaré series
scientific article; zbMATH DE number 5561729
scientific article