REVIEW 2 major objections 1 minor 1 cited by
Decomposition Theorem for Perfectoid Rings along General Ideals
T0 review · 2 major / 1 minor · reviewed 2026-07-12 · grok-4.5
Pith's one-line read In a perfectoid ring R, the I-torsion submodule is almost zero with respect to the perfectoidization of any ideal I, yielding an excision-type decomposition of R along that torsion.
desk verdict Abstract-only structural result on I-torsion tameness in perfectoid rings; plausible outline, but scope for general ideals is unchecked. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The excision square for perfectoidization supplied by p-complete arc descent, combined with André’s lemma; together they control the almost-vanishing of torsion and produce the decomposition of R.
What would settle it
Exhibit a perfectoid ring R and an ideal I for which some nonzero I-torsion element of R fails to be annihilated by every element of a power of I_perfd, or for which the corresponding excision square of perfectoidizations fails to be Cartesian.
Extended reading notes
Core claim
If R is a perfectoid ring and I is an arbitrary ideal of R, then the I-torsion submodule of R is I_perfd-almost zero; consequently R admits an excision-type decomposition that isolates its I-torsion part via the perfectoidization of I.
Load-bearing premise
The excision square for perfectoidization coming from p-complete arc descent must remain valid for completely general ideals, not only for special or finitely generated ones.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims a tameness theorem for torsion in perfectoid rings: if R is perfectoid and I any ideal, the I-torsion submodule of R is I_perfd-almost zero. The argument is said to combine André’s lemma with the excision square for perfectoidization arising from p-complete arc descent, and the almost-vanishing is then used to produce an excision-type decomposition of R along its I-torsion part. Additional structural remarks on (semi)perfectoid rings and perfectoid ideals are announced.
Significance. A decomposition of an arbitrary perfectoid ring along the torsion with respect to a completely general ideal would be a useful structural tool, extending known almost-mathematics results that typically impose finite-generation or other restrictions on I. The named ingredients (André’s lemma, p-complete arc descent) are standard and appropriate; if the claimed generality holds, the paper supplies a clean, parameter-free tameness statement that subsequent work on perfectoidization and prismatic cohomology could cite.
major comments (2)
- [Abstract] The abstract asserts the tameness statement for an arbitrary ideal I. The only indicated tools are André’s lemma and the excision square coming from p-complete arc descent. Without the body of the paper it is impossible to verify whether that square is established (or even stated) for non-finitely-generated ideals; if the square requires finite generation, perfectoidness of I, or other unstated hypotheses, the main claim fails in the advertised generality. This is a load-bearing scope issue.
- [Abstract] The abstract gives no indication of the precise almost-zero estimates or of the intermediate lemmas that convert the excision square into the statement that I-torsion is I_perfd-almost zero. Those estimates are essential for the subsequent decomposition; their absence from the available text leaves the central derivation unchecked.
minor comments (1)
- [Abstract] The abstract is clear and self-contained as far as it goes, but the phrase “excision-type decomposition” is left undefined; a one-sentence expansion of what the resulting Cartesian square or short exact sequence looks like would help readers assess the strength of the claim.
Circularity Check
No significant circularity detectable; abstract presents a theorem deduced from André's lemma and p-complete arc descent without definitional or fitted reductions.
full rationale
Only the abstract is available. It states that the main tameness theorem (I-torsion in a perfectoid ring R is I_perfd-almost zero) and the resulting excision-type decomposition are proved using André's lemma and the excision square for perfectoidization arising from p-complete arc descent. André's lemma is an external foundational result (not by the present authors). No equations, fitted parameters, uniqueness theorems imported from the authors' prior work, or self-definitional steps appear in the abstract. There is therefore no quotable reduction of a claimed prediction or first-principles result to its own inputs by construction. Dependence on standard tools of perfectoid theory is ordinary mathematical practice and does not constitute circularity under the stated criteria. Score 0 is the honest finding for an abstract-only pure-math paper whose outline is coherent and non-circular on its face.
Assumptions & free parameters
assumptions (4)
- domain assumption André's lemma (as used for perfectoid rings / perfectoidization)
- domain assumption Excision square for perfectoidization from p-complete arc descent
- domain assumption Standard definition and basic properties of perfectoid rings and perfectoidization
- domain assumption Almost mathematics / almost-zero formalism relative to I_perfd
Cite this review
Pith. "Pith review of Decomposition Theorem for Perfectoid Rings along General Ideals." pith.science (2026). https://pith.science/paper/Y3XGKZS5
@misc{pith2026260606241,
author = {Pith},
title = {Pith review of: Decomposition Theorem for Perfectoid Rings along General Ideals},
year = {2026},
howpublished = {\url{https://pith.science/paper/Y3XGKZS5}},
note = {Machine review of arXiv:2606.06241}
}
abstract
Using Andr\'e's lemma and the excision square for perfectoidization coming from $p$-complete arc descent, we prove new structural results about perfectoid rings and perfectoidization. The main result is a tameness theorem for torsion in perfectoid rings: if $R$ is a perfectoid ring and $I\subset R$ is an ideal, then the $I$-torsion in $R$ is $I_{\mathrm{perfd}}$-almost zero. This yields an excision-type decomposition of $R$ along its $I$-torsion part. We also study (semi)perfectoid rings and perfectoid ideals and take the opportunity to make some structural remarks about them.
Forward citations
Cited by 1 Pith paper
-
On Perfectoidizaiton of Finite Algebras over a Perfectoid Ring
Proves d A_pfd subset A for monic polynomials with bounded-torsion discriminant over perfectoid R, plus density criterion reducing to p-power roots mod p, and computes examples in Kummer and split cases.
Reviewed July 12, 2026 · model on record in the stance chip above.
Discussion (0). Sign in to comment.