The Proj of the Rees algebra of a graded family of ideals
Steven Dale Cutkosky

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 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…
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