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arxiv: 2208.00051 · v3 · pith:JBLQMZSPnew · submitted 2022-07-29 · 🧮 math.AC · math.AG

Diagonal F-splitting and Symbolic Powers of Ideals

Daniel Smolkin This is my paper

Pith reviewed 2026-05-24 11:22 UTC · model grok-4.3

classification 🧮 math.AC math.AG
keywords diagonal F-splittingtest idealssymbolic powersstrongly F-regular ringsdeterminantal ringstoric ringspositive characteristicideal containments
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The pith

In diagonally F-split rings, J^{s+t} is contained in the product of test ideals τ(J^{s-ε}) τ(J^{t-ε}) for any ideal J.

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

The paper proves that in a strongly F-regular ring that is also diagonally F-split and essentially of finite type over an F-finite field, ordinary powers of an ideal sit inside a product of its test ideals. This containment is then applied to prime ideals to produce relations between their symbolic powers and ordinary powers. The relations are new in this setting and cover all determinantal rings together with many toric rings in positive characteristic. In the strongest case the paper obtains an explicit numerical bound: the 2hn-th symbolic power of a height-h prime is contained in its n-th ordinary power. A reader cares because symbolic powers govern the geometry of varieties while ordinary powers are the ones that can be computed directly from generators.

Core claim

Let R be a strongly F-regular, diagonally F-split ring essentially of finite type over an F-finite field. For any ideal J and positive real numbers s, t, ε for which the expression is defined, the ordinary power J^{s+t} is contained in the product τ(J^{s-ε}) τ(J^{t-ε}) of test ideals. The diagonal F-splitting supplies the key map that produces this product containment. The same argument yields containments between symbolic and ordinary powers of primes; the sharpest explicit form is P^{(2 h n)} ⊆ P^n for every prime ideal P of height h.

What carries the argument

Diagonal F-splitting, a ring homomorphism from R to the tensor product R ⊗_R R that splits the multiplication map along the diagonal and is used to manufacture the test-ideal product containment.

If this is right

  • The containment J^{s+t} ⊆ τ(J^{s-ε}) τ(J^{t-ε}) holds for every ideal J in the stated rings.
  • P^{(2 h n)} ⊆ P^n holds for every prime ideal P of height h.
  • The same numerical containments hold in every determinantal ring and in a large class of toric rings in positive characteristic.
  • Symbolic-versus-ordinary power relations are obtained for arbitrary primes rather than only for special classes.

Where Pith is reading between the lines

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

  • If diagonal F-splitting turns out to hold for a wider class of strongly F-regular rings, the same power containments would apply more broadly.
  • The numerical bound 2 h n may be compared with known uniform symbolic power bounds in other characteristics to test sharpness.
  • The result supplies a concrete test for whether a given ring is diagonally F-split by checking whether the power containments fail.

Load-bearing premise

The ring must admit a diagonal F-splitting in addition to being strongly F-regular.

What would settle it

An explicit diagonally F-split ring R, ideal J, and numbers s, t, ε > 0 such that J^{s+t} is not contained in τ(J^{s-ε}) τ(J^{t-ε}), or a height-h prime P with P^{(2 h n)} not contained in P^n.

read the original abstract

Let $J$ be any ideal in a strongly $F$-regular, diagonally $F$-split ring $R$ essentially of finite type over an $F$-finite field. We show that $J^{s+t} \subseteq \tau(J^{s - \epsilon}) \tau(J^{t-\epsilon})$ for all $s, t, \epsilon > 0$ for which the formula makes sense. We use this to show a number of novel containments between symbolic and ordinary powers of prime ideals in this setting, which includes all determinantal rings and a large class of toric rings in positive characteristic. In particular, we show that $P^{(2hn)} \subseteq P^n$ for all prime ideals $P$ of height $h$ in such rings.

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.

Referee Report

0 major / 3 minor

Summary. The paper proves that if R is a strongly F-regular, diagonally F-split ring essentially of finite type over an F-finite field, then for any ideal J the containment J^{s+t} ⊆ τ(J^{s−ε}) τ(J^{t−ε}) holds for all s, t, ε > 0 where the expression is defined. This test-ideal product is then applied to obtain new symbolic-versus-ordinary power containments for prime ideals, including the uniform bound P^{(2hn)} ⊆ P^n for every prime P of height h; the results apply in particular to determinantal rings and a large class of toric rings in positive characteristic.

Significance. If the central containment holds, the work supplies the first uniform symbolic-power bounds of this form that are valid in all determinantal rings and in many toric rings, by leveraging the additional diagonal F-splitting hypothesis on top of strong F-regularity. The derivation is presented as a direct argument from the splitting assumption rather than a reduction to fitted quantities, and the paper explicitly lists the rings to which the hypothesis applies.

minor comments (3)
  1. The precise statement of the diagonal F-splitting hypothesis (Definition 2.3 or equivalent) should be recalled verbatim in the introduction so that readers can immediately see the extra assumption beyond strong F-regularity.
  2. In the proof of the main containment (likely §3), the single use of the diagonal splitting map should be isolated in a separate displayed step or lemma to make the dependence on the hypothesis transparent.
  3. The paper should include a short remark comparing the new bound P^{(2hn)} ⊆ P^n with the best previously known exponents in the strongly F-regular case alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive summary of the results, and recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from diagonal F-split hypothesis

full rationale

The paper derives the test-ideal product containment J^{s+t} ⊆ τ(J^{s-ε})τ(J^{t-ε}) directly from the assumption that R is diagonally F-split (plus strong F-regularity and finite-type hypotheses). This containment is then used to obtain the symbolic-power results such as P^{(2hn)} ⊆ P^n. No load-bearing step reduces by definition, by fitting a parameter, or by a self-citation chain to the target conclusion; the diagonal-splitting property is an independent hypothesis whose verification for determinantal and toric rings is stated separately. The argument is therefore a standard direct proof in the restricted setting and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the standard definitions of strong F-regularity, diagonal F-splitting, test ideals, and symbolic powers in positive-characteristic commutative algebra; no free parameters or new entities are introduced.

axioms (3)
  • domain assumption R is strongly F-regular
    Invoked in the statement of the main theorem as the ambient ring class.
  • domain assumption R is diagonally F-split
    The key technical hypothesis used to obtain the test-ideal product containment.
  • domain assumption R is essentially of finite type over an F-finite field
    Standard setup for F-finite positive-characteristic rings.

pith-pipeline@v0.9.0 · 5652 in / 1461 out tokens · 25548 ms · 2026-05-24T11:22:55.246426+00:00 · methodology

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