On $p_g$-ideals

TL;DR

This paper extends the existence and stability of $p_g$-ideals, originally proven over algebraically closed fields, to excellent normal domains of dimension two with characteristic zero, showing their product remains a $p_g$-ideal.

Contribution

The paper proves that in characteristic zero, excellent normal domains of dimension two contain $p_g$-ideals and their products are also $p_g$-ideals, generalizing previous results.

Findings

Existence of $p_g$-ideals in characteristic zero

Product of two $p_g$-ideals is a $p_g$-ideal

Extension of previous results to new class of rings

Abstract

Let $(A,\mathfrak{m})$ be an excellent normal domain of dimension two. We define an $\mathfrak{m}$-primary ideal $I$ to be a $p_g$-ideal if the Rees algebra $A[It]$ is a Cohen-Macaulay normal domain. When $A$ contains an algebraically closed field $k \cong A/\mathfrak{m}$ then Okuma, Watanabe and Yoshida proved that $A$ has $p_g$-ideals and furthermore product of two $p_g$-ideals is a $p_g$ ideal. In this article we show that if $A$ is an excellent normal domain of dimension two containing a field $k \cong A/\mathfrak{m}$ of characteristic zero then also $A$ has $p_g$-ideals. Furthermore product of two $p_g$-ideals is $p_g$.