The Proj of the Rees algebra of a graded family of ideals

TL;DR

This paper studies when the Proj of Rees algebras of graded ideal families in Noetherian local rings is Noetherian, revealing dimension-dependent behaviors and providing examples of non-Noetherian cases.

Contribution

It extends understanding of the conditions under which the Proj of Rees algebras is Noetherian, especially in relation to divisorial filtrations and dimension.

Findings

Proj of Rees algebra is always Noetherian for zero analytic spread.

In two-dimensional rings, the Proj of divisorial filtrations is always Noetherian.

Counterexamples show non-Noetherian Proj in higher dimensions.

Abstract

In this article we investigate the condition that the Proj of a Rees algebra of a graded family of ideals in a Noetherian local ring $R$ is Noetherian. In many cases, the Proj will be Noetherian even when the Rees algebra is not. For instance, the Proj of the Rees algebra of a graded filtration of ideals will alway be Noetherian if the analytic spread of the filtration is zero. The Proj of a Rees algebra of a divisorial filtration on a two dimensional normal excellent local ring is always Noetherian, as was proven by Russo and later with a different proof by the author. We give examples in this paper of divisorial filtrations on three dimensional normal excellent local rings whose Proj is not Noetherian, showing that this theorem does not extend to higher dimensions. A consequence of the fact that the Proj of a divisorial filtration over a two dimensional excellent normal local ring is always Noetherian is that the preimage of the maximal ideal of $R$ in the Proj has only finitely many irreducible components. As a consequence, the fiber cone of such a filtration has only finitely many minimal primes. We give an example of a graded filtration of ideals in a two dimensional regular local ring such that the preimage of the maximal ideal in the Proj of the Rees algebra of the filtration has infinitely many irreducible components, so that the Proj is not Noetherian, and the fiber cone of the filtration has infinitely many minimal primes.