The Proj of the Rees algebra of a graded family of ideals
Pith reviewed 2026-05-18 05:23 UTC · model grok-4.3
The pith
Proj of the Rees algebra of a divisorial filtration need not be Noetherian in three dimensions.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The Proj of the Rees algebra of a divisorial filtration on a two dimensional normal excellent local ring is always Noetherian. Examples exist of divisorial filtrations on three dimensional normal excellent local rings whose Proj is not Noetherian. In the two-dimensional case the preimage of the maximal ideal therefore has only finitely many irreducible components and the fiber cone has only finitely many minimal primes, but a graded filtration on a two-dimensional regular local ring can be constructed so that this preimage has infinitely many irreducible components.
What carries the argument
Proj of the Rees algebra of a graded family of ideals, especially when the family is a divisorial filtration on a normal local ring.
If this is right
- When the analytic spread of a graded filtration is zero, its Rees algebra has Noetherian Proj.
- In two dimensions the Noetherian property of the Proj implies the fiber cone has finitely many minimal primes.
- The preimage of the maximal ideal in the Proj has only finitely many irreducible components when the filtration is divisorial in two dimensions.
- These finiteness conclusions fail for the constructed examples in three dimensions and for the special filtration in two dimensions.
Where Pith is reading between the lines
- Dimension appears to control whether the Proj of a divisorial filtration must be Noetherian.
- One could seek a precise dimension threshold beyond which such non-Noetherian examples become possible.
- Analogous questions arise for other natural classes of filtrations beyond the divisorial case.
Load-bearing premise
The counterexamples in three dimensions depend on the existence of particular graded divisorial filtrations whose Proj is non-Noetherian.
What would settle it
Verification in one of the concrete three-dimensional examples that the Proj is in fact Noetherian would show the claimed counterexamples do not work.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper investigates when the Proj of the Rees algebra of a graded family of ideals in a Noetherian local ring is Noetherian. It observes that the Proj is always Noetherian if the analytic spread of the filtration is zero. It recalls the theorem (due to Russo and the author) that the Proj of the Rees algebra of a divisorial filtration on a two-dimensional normal excellent local ring is always Noetherian. The central contribution consists of explicit constructions of divisorial filtrations on three-dimensional normal excellent local rings for which the Proj is not Noetherian. As a contrasting illustration, the paper also gives an example of a graded filtration on a two-dimensional regular local ring whose Proj has infinitely many irreducible components over the maximal ideal, so that the fiber cone has infinitely many minimal primes.
Significance. The counterexamples demonstrate that the two-dimensional Noetherian result for Proj of Rees algebras of divisorial filtrations does not extend to dimension three. This is a substantive clarification of the dimensional dependence of the property. The explicit constructions of the filtrations and the verification that they are divisorial constitute a strength, as they permit direct checking of the non-Noetherian conclusion via the fiber cone having infinitely many minimal primes. The two-dimensional non-divisorial example usefully isolates the role of the divisorial hypothesis.
major comments (1)
- §4 (three-dimensional counterexample construction): the argument that the chosen graded family is divisorial and that the associated Proj has infinitely many irreducible components over the maximal ideal (equivalently, that the fiber cone has infinitely many minimal primes) is load-bearing for the claim that the two-dimensional theorem fails in dimension three. The verification that the ring remains normal and excellent and that no hidden relations collapse the components should be expanded with an explicit computation of the relevant valuations or associated primes to rule out post-hoc normalization effects.
minor comments (2)
- Abstract: 'alway' should read 'always'.
- Introduction: the citation to Russo's result would benefit from the precise reference and a one-sentence summary of the proof strategy for context.
Simulated Author's Rebuttal
We thank the referee for the careful reading of the manuscript and for the constructive comments. We address the major comment below and will incorporate the suggested expansions in the revised version.
read point-by-point responses
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Referee: §4 (three-dimensional counterexample construction): the argument that the chosen graded family is divisorial and that the associated Proj has infinitely many irreducible components over the maximal ideal (equivalently, that the fiber cone has infinitely many minimal primes) is load-bearing for the claim that the two-dimensional theorem fails in dimension three. The verification that the ring remains normal and excellent and that no hidden relations collapse the components should be expanded with an explicit computation of the relevant valuations or associated primes to rule out post-hoc normalization effects.
Authors: We agree that the verification in §4 is load-bearing and that additional explicit computations would strengthen the presentation. In the revised manuscript we will expand this section with detailed calculations of the relevant valuations (including the orders along the exceptional divisors in the blow-up) and the associated primes of the fiber cone. These computations will explicitly confirm that the ring remains normal and excellent, that the filtration is divisorial, and that no hidden relations collapse the infinitely many minimal primes of the fiber cone. We believe the current outline already rules out post-hoc normalization effects, but the added details will make this fully transparent. revision: yes
Circularity Check
Minor self-citation to prior 2D result provides context but is not load-bearing for new 3D counterexamples
full rationale
The paper references Russo's proof and the author's own prior proof that the Proj of the Rees algebra of a divisorial filtration is Noetherian in the 2D normal excellent local ring case. This is standard self-citation for background and does not reduce the central new claims (explicit 3D divisorial filtration counterexamples where Proj fails to be Noetherian, plus a 2D non-divisorial example with infinitely many irreducible components) to a self-referential fit or definition. The constructions rest on standard definitions of Rees algebras, Proj, divisorial filtrations, and analytic spread, with the non-Noetherian property following from the specific graded families chosen rather than from the cited 2D theorem by construction. No self-definitional steps, fitted inputs renamed as predictions, or ansatz smuggling appear in the abstract or described claims. The derivation chain remains self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption R is a Noetherian local ring
- domain assumption Filtrations are graded families of ideals (or divisorial filtrations)
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/AbsoluteFloorClosure.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
The Proj of a Rees algebra of a divisorial filtration on a two dimensional normal excellent local ring is always Noetherian... We give examples... of divisorial filtrations on three dimensional normal excellent local rings whose Proj is not Noetherian.
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
discussion (0)
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