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

Steven Dale Cutkosky

TL;DR

This paper investigates when Proj$(R[\mathcal{I}])$ is Noetherian for a graded family of ideals in a Noetherian local ring, focusing on divisorial filtrations. It shows that in dimension two, Proj$(R[\mathcal{I}])$ is Noetherian for divisorial filtrations and analyzes the fiber cone structure, while in dimension three there are counterexamples with non-Noetherian Proj, underscoring a dimensional boundary. The authors develop valuation-based invariants $\gamma_v(\mathcal{I})$, generalize McAdam-type results via resolutions/alterations, and construct filtrations with fiber cones having infinitely many irreducible components, as well as symbolic powers with infinitely many Rees valuations. They employ Beatty sequences to demonstrate oscillatory behavior in growth rates and establish both positive and pathological phenomena in Hilbert functions, highlighting the intricate interplay between valuations, filtrations, and birational geometry. Overall, the work delineates when Noetherian Proj behavior holds and provides tools (valuations, alterations, and analytic-spread analysis) to study asymptotic properties of filtrations 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.

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

TL;DR

This paper investigates when Proj is Noetherian for a graded family of ideals in a Noetherian local ring, focusing on divisorial filtrations. It shows that in dimension two, Proj is Noetherian for divisorial filtrations and analyzes the fiber cone structure, while in dimension three there are counterexamples with non-Noetherian Proj, underscoring a dimensional boundary. The authors develop valuation-based invariants , generalize McAdam-type results via resolutions/alterations, and construct filtrations with fiber cones having infinitely many irreducible components, as well as symbolic powers with infinitely many Rees valuations. They employ Beatty sequences to demonstrate oscillatory behavior in growth rates and establish both positive and pathological phenomena in Hilbert functions, highlighting the intricate interplay between valuations, filtrations, and birational geometry. Overall, the work delineates when Noetherian Proj behavior holds and provides tools (valuations, alterations, and analytic-spread analysis) to study asymptotic properties of filtrations 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 is always Noetherian is that the preimage of the maximal ideal of 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.
Paper Structure (30 sections, 31 theorems, 294 equations)

This paper contains 30 sections, 31 theorems, 294 equations.

Table of Contents

  1. Introduction
  2. A divisorial filtration $\mathcal{I}$ on a 3 dimensional normal excellent local ring such that $\hbox{Proj}(R[\mathcal{I}])$ is not Noetherian.
  3. an invariant
  4. A graded filtration whose fiber cone has infinitely many irreducible components
  5. Calculation of a family of divisors
  6. Construction of the example
  7. Introduction
  8. Graded Filtrations, Rees Algebras and Analytic Spread
  9. McAdam's theorem for divisorial filtrations
  10. Hilbert functions of divisorial $m_R$-filtrations
  11. Symbolic powers with infinitely many Rees valuations
  12. Acknowledgements
  13. Valuations, Filtrations and Rees Algebras
  14. Extensions of valuations and Rees algebras
  15. Divisorial Filtrations
  16. ...and 15 more sections

Key Result

Theorem 1.1

(Theorem 4 R, Theorem 1.3 C1) Suppose that $R$ is a 2-dimensional normal excellent local ring and $\mathcal{I}$ is a divisorial filtration on $R$. Then $\hbox{Proj}(R[\mathcal{I}])$ is a Noetherian normal scheme. It is proper over $\hbox{Spec}(R)$ if and only if $R[\mathcal{I}]$ is Noetherian.

Theorems & Definitions (54)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • Proposition 3.1
  • ...and 44 more