Un exemple effectif de gradué non noethérien associé à une valuation divisorielle. (An effective example of a non noetherian graduate ring associated to a divisorial valuation)
For a local regular ring \(R\) and with the cubic \(x^2z+xy^2+y^3\) the authors construct a sequence of blow-ups \(\text{Spec }R=X_0\leftarrow X_1\leftarrow\dots\leftarrow X_n\), \(n\geq 10\) and a divisorial valuation \(\nu\), so that, for \(I_\alpha=\{ r\in R\mid \nu(r)\geq\alpha\}\), \(\text{gr}_\nu(R)=\bigoplus I_\alpha/I_{\alpha+1}\) is not Noetherian (for \(n\leq 9\), \(\text{gr}_\nu(R)\) is Noetherian). They also prove that the Poincaré series of \(\text{gr}_\nu(R)\) is a rational fraction.
scientific article; zbMATH DE number 1409065
scientific article