Integrality over ideal semifiltrations

Darij Grinberg

arxiv: 1907.06125 · v1 · pith:Z23PVAE5new · submitted 2019-07-13 · 🧮 math.AC · math.RA

Integrality over ideal semifiltrations

Darij Grinberg This is my paper

Pith reviewed 2026-05-24 21:55 UTC · model grok-4.3

classification 🧮 math.AC math.RA

keywords integralityideal semifiltrationcommutative algebraRees algebraintegral closuretransitivityA-algebramultiplicative closure

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The pith

If u is integral over both A[x] and A[y] in a commutative A-algebra, then u is integral over A[xy].

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the classical notion of integrality over rings to integrality over ideal semifiltrations, which are sequences of ideals starting with the base ring and closed under the rule that the product of two ideals lands in the ideal indexed by the sum of their indices. It first recalls standard results on transitivity and closure under sums and products for ordinary integrality, then gives a criterion that reduces the semifiltration version to ordinary integrality over a ring constructed from a variant of the Rees algebra. With this reduction in hand, the same closure properties are shown to hold for the generalized notion. The central new fact is that an element integral over the subring generated by x and also over the subring generated by y must be integral over the subring generated by their product xy, and this fact lifts to the semifiltration setting.

Core claim

Integrality over an ideal semifiltration reduces to ordinary integrality over a ring via a variant of the Rees algebra. This reduction is used to prove that the generalized integrality is transitive and closed under sums and products. In particular, if u, x, and y are elements of a commutative A-algebra such that u is integral over A[x] and integral over A[y], then u is integral over A[xy]; the same conclusion holds when integrality is taken with respect to ideal semifiltrations instead of single subrings.

What carries the argument

The ideal semifiltration, a sequence of ideals (I0 = A, I1, I2, ...) satisfying Ia Ib ⊆ I_{a+b} for all a, b, which generalizes both ring integrality and ideal integrality, together with the reduction criterion that converts questions about it into ordinary ring integrality via a Rees-algebra variant.

If this is right

Transitivity of integrality continues to hold when the base is an ideal semifiltration rather than a single ring.
If two elements are integral over an ideal semifiltration, then their sum and product are also integral over that semifiltration.
The reduction criterion supplies an explicit computational test for the generalized integrality.
The closure property for products xy lifts directly from the ring case to the semifiltration case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result may simplify explicit calculations of integral closures when the algebra is generated by several elements whose product appears in relations.
The same reduction technique could be applied to other multiplicative sequences of ideals that arise in the study of filtrations on rings.
Concrete verification could begin by testing the A[xy] claim inside polynomial rings over the integers or over finite fields.

Load-bearing premise

That integrality over an ideal semifiltration can be reduced to ordinary integrality over a ring by means of a variant of the Rees algebra.

What would settle it

A concrete commutative A-algebra containing elements u, x, y such that u satisfies a monic equation over A[x] and over A[y] but fails to satisfy any monic equation over A[xy].

read the original abstract

We study integrality over rings (all commutative in this paper) and over ideal semifiltrations (a generalization of integrality over ideals). We begin by reproving classical results, such as a version of the "faithful module" criterion for integrality over a ring, the transitivity of integrality, and the theorem that sums and products of integral elements are again integral. Then, we define the notion of integrality over an ideal semifiltration (a sequence $\left( I_0,I_1,I_2,\ldots\right)$ of ideals satisfying $I_0 =A$ and $I_a I_b \subseteq I_{a+b}$ for all $a,b\in\mathbb{N}$), which generalizes both integrality over a ring and integrality over an ideal (as considered, e.g., in Swanson/Huneke, "Integral Closure of Ideals, Rings, and Modules"). We prove a criterion that reduces this general notion to integrality over a ring using a variant of the Rees algebra. Using this criterion, we study this notion further and obtain transitivity and closedness under sums and products for it as well. Finally, we prove the curious fact that if $u$, $x$ and $y$ are three elements of a (commutative) $A$-algebra (for $A$ a ring) such that $u$ is both integral over $A\left[ x\right]$ and integral over $A\left[ y\right]$, then $u$ is integral over $A\left[ xy\right]$. We generalize this to integrality over ideal semifiltrations, too.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Paper defines integrality over ideal semifiltrations and proves the A[xy] generalization, with the reduction via Rees algebra variant looking internally consistent.

read the letter

The main thing to know is that this paper defines integrality over ideal semifiltrations and uses the definition to generalize the statement that integrality over A[x] and over A[y] implies integrality over A[xy]. It also recovers the usual transitivity and sum/product properties in the new setting. The approach begins with reproofs of the classical faithful module criterion, transitivity, and closure under sums and products for ordinary integrality. These are set up to carry over once the semifiltration case is reduced to the ring case. The reduction itself uses a variant of the Rees algebra built from the given axioms (I_0 = A and I_a I_b ⊆ I_{a+b}), which is a standard move in this area and recovers the known special cases of ring integrality and ideal integrality. The A[xy] fact is presented as a curious observation and then lifted to the semifiltration level; the argument is direct once the reduction is in hand. The stress-test note that the construction is internally consistent and avoids circularity matches the abstract. No load-bearing gaps are visible from the description. The paper stays strictly within commutative algebra and does not claim downstream applications or new examples outside the integral-closure literature. That keeps the scope limited but also keeps the claims precise. The citation to Swanson/Huneke is appropriate and the work builds on it without overclaiming. This is for readers already working on integral dependence, Rees algebras, or ideal filtrations. Someone who knows the classical criteria will see the extension without much extra effort. It shows clear, honest engagement with the definitions and reductions. I would send it to peer review; the new notion and the generalized theorem are substantive enough to justify referee time even if the proofs need polishing for readability.

Referee Report

0 major / 3 minor

Summary. The manuscript studies integrality over commutative rings, reproving classical results such as the faithful module criterion, transitivity, and closure under sums and products. It defines integrality over ideal semifiltrations (sequences of ideals (I_0, I_1, ...) with I_0 = A and I_a I_b ⊆ I_{a+b}), proves a criterion reducing this to ordinary ring integrality via a Rees algebra variant, extends the classical properties to the new setting, and establishes that if u is integral over A[x] and over A[y] then u is integral over A[xy], with a corresponding generalization to ideal semifiltrations.

Significance. If the results hold, the work supplies a unified treatment of integrality that recovers ordinary ring integrality and ideal integrality as special cases. The reduction via the Rees-algebra variant is a useful technical tool that permits direct transfer of known theorems, while the A[x], A[y] → A[xy] statement is a non-obvious closure property that may find applications in the study of integral closures. The self-contained reproofs of the classical facts enhance readability and make the paper a self-contained reference.

minor comments (3)

[Abstract] Abstract: the statement that the reduction 'uses a variant of the Rees algebra' is central but left at a high level; a one-sentence indication of the construction (e.g., the precise grading or the module used) would help readers locate the key step.
The reference to Swanson/Huneke is appropriate but should appear with full bibliographic details (edition, year, publisher) in the bibliography section.
Notation: the indexing set for the semifiltration is stated as ℕ in the abstract; confirm whether 0 is included and whether the sequence is required to be exhaustive or only non-decreasing in the formal definition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and recommendation of minor revision. No specific major comments were provided in the report, so we have no points requiring rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper explicitly reproves classical integrality criteria (faithful module, transitivity, sums/products) from first principles before introducing the ideal semifiltration definition and its reduction to ordinary ring integrality via an explicit Rees-algebra variant. All subsequent results, including the A[x]/A[y] to A[xy] claim and its generalization, are derived from this reduction and the reproved standard facts without any fitted parameters, self-referential equations, or load-bearing self-citations. The construction begins from the stated semifiltration axioms and recovers known cases as special instances, making the chain independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper introduces definitions within standard commutative algebra and relies on classical results without new free parameters or postulated entities.

axioms (2)

domain assumption All rings and algebras considered are commutative
Explicitly stated in the abstract.
standard math Standard properties of ideals, polynomial rings, and the Rees algebra construction hold
Invoked when reducing integrality over semifiltrations to ring integrality.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean; AlexanderDuality.lean reality_from_one_distinction; alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

ideal semifiltration (I₀=A and IₐI_b⊆I_{a+b})

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