On the prime ideal structure of symbolic Rees algebras

TL;DR

This paper explores the prime ideal structure and dimension theory of symbolic Rees algebras over Noetherian domains, providing new insights into their properties and applications, including counterexamples and conditions for ideal power coincidences.

Contribution

It establishes general results on prime spectra of subalgebras between a domain and its localization, and analyzes the Noetherian-like behavior of symbolic Rees algebras, including special cases and applications.

Findings

Rees algebra of a maximal ideal inherits stable strong S-domain properties.

Counterexample to the Zariski-Hilbert problem using symbolic Rees algebras.

Conditions under which symbolic and ordinary powers of a prime ideal coincide.

Abstract

This paper contributes to the study of the prime spectrum and dimension theory of symbolic Rees algebra over Noetherian domains. We first establish some general results on the prime ideal structure of subalgebras of affine domains, which actually arise, in the Noetherian context, as domains between a domain $A$ and $A[a^{-1}]$. We then examine closely the special context of symbolic Rees algebras (which yielded the first counter-example to the Zariski-Hilbert problem). One of the results states that if $A$ is a Noetherian domain and $p$ a maximal ideal of $A$, then the Rees algebra of $p$ inherits the Noetherian-like behavior of being a stably strong S-domain. We also investigate graded rings associated with symbolic Rees algebras of prime ideals $p$ such that $A_{p}$ is a rank-one DVR and close with an application related to Hochster's result on the coincidence of the ordinary and symbolic powers of a prime ideal.