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On Perfectoidizaiton of Finite Algebras over a Perfectoid Ring

T0 review · 2 major / 7 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read Perfectoidization of monic finite algebras over a perfectoid ring is controlled by the discriminant: d times the perfectoidization sits inside the original algebra.

desk verdict Solid specialist paper: sharp monogenic discriminant bound d A_perfd ⊆ A plus explicit Kummer/split models that people will actually use. read the letter →

arxiv 2606.12229 v2 pith:I3VYAE7L submitted 2026-06-10 math.AC math.AGmath.NT

Ryo Ishizuka , Léo Navarro Chafloque This is my paper · ORCID
classification math.ACmath.AGmath.NT MSC 14G4513J10
keywords perfectoidringsperfectoidizationdiscriminantKummerextensionsboundedtorsionp-rootclosurefinitealgebrasalmostpurity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies what happens when you take a finite algebra A over a perfectoid ring R and form its perfectoidization A_perfd, the initial perfectoid A-algebra. In the monogenic case A = R[t]/(m(t)) with monic m, the discriminant d of m kills the cone of A to A_perfd. When d is a nonzerodivisor and R/dR has bounded p-power torsion, A_perfd embeds into A[1/d] and satisfies the concrete containment d A_perfd subset A. A density criterion further shows that adjoining enough p-power roots modulo p is enough to recover the full perfectoidization after p-completion. Explicit computations then identify the perfectoidizations of Kummer extensions R[t]/(t^m - r) and of split monic polynomials, often as rings of functions that are constant modulo a perfectoid ideal. The results give a precise description of how much new material is forced by perfectoidization and when that material remains “almost finite” over the original algebra.

What carries the argument

The discriminant d = Res(m, m') together with the density principle: an intermediate algebra C with C/pC semiperfect is dense in A_perfd for the p-adic topology, and equals it after p-completion when V(d)=V(p).

What would settle it

Exhibit a monic polynomial over a perfectoid ring whose discriminant d is a nonzerodivisor with unbounded p-power torsion on R/dR, such that some element of A_perfd is not of the form a/d with a in A.

Watch

Extended reading notes

Core claim

For R perfectoid and A = R[t]/(m(t)) monic, the cone of A to its perfectoidization is killed by the discriminant d of m. When d is a nonzerodivisor and R/dR has bounded p^infty-torsion, A_perfd is d-torsion-free, sits inside A[1/d], and satisfies d A_perfd subset A.

Load-bearing premise

The bounded p-power torsion condition on R/dR, without which d may cease to be a nonzerodivisor on the perfectoidization and the embedding into A[1/d] can fail.

Editorial extensions

If this is right

  • Any element of the perfectoidization of a monogenic finite algebra can be written as a fraction with denominator the discriminant.
  • Kummer extensions R[t]/(t^m-r) with (m,p)=1 have perfectoidization obtained by adjoining all compatible p-power roots of the m-th root of r (or of a unit twist of a p-power root of a uniformizer).
  • Split monic polynomials yield perfectoidizations that are rings of functions constant modulo the perfectoidization of (d).
  • p-root closure of A inside A[1/p] coincides with A_perfd when A is p-torsion-free and finite étale after inverting p.
  • The same descriptions give the perfectoidization of certain semiperfectoid rings as continuous almost-constant functions on profinite sets such as Z_p.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The discriminant control may extend to finite locally free algebras once a suitable discriminant ideal is defined, recovering the same “hidden finiteness.”
  • The density criterion suggests an algorithmic path: generate enough p-power roots modulo p and complete, rather than construct the whole initial object abstractly.
  • Arc-local descriptions of Kummer perfectoidizations give a practical way to test almost-purity statements by base change to perfectoid valuation rings.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. The paper studies the perfectoidization A_perfd of finite finitely presented algebras A over a perfectoid ring R. The main theoretical results are: (i) for monogenic A = R[t]/(m(t)) with monic m and discriminant d a nonzerodivisor such that R/dR has bounded p^∞-torsion, Cone(A → A_perfd) is killed by d, so A_perfd embeds into A[1/d] and d A_perfd ⊆ A (Theorem 2.11); (ii) a density criterion: an intermediate A ⊆ C ⊆ A_perfd with C/pC semiperfect is dense in the p-adic topology on A_perfd, and equals A_perfd after p-completion when V(d)=V(p) (Theorem 2.15 / Corollary 2.16); (iii) a p-root-closure description of A_perfd when R and A are p-torsion-free and the map is étale after inverting p (Proposition 2.13). Section 3 computes A_perfd explicitly for Kummer extensions R[t]/(t^m − r) (Theorems 3.1, 3.6, Corollary 3.4), for split monic polynomials via the ring Fun_{d-acst}(n,R) of functions constant modulo (d)_perfd (Theorem 3.9 and examples), and for certain semiperfectoids via continuous almost-constant functions on Z_p (Proposition 3.18).

Significance. If the results hold, the paper supplies concrete control and explicit models for perfectoidizations of finite algebras, which are typically only known to exist abstractly via Bhatt–Scholze. The monogenic containment d A_perfd ⊆ A is a useful “hidden finiteness” statement; the density criterion reduces constructions to adjoining p-power roots modulo p; and the Kummer and split computations give usable descriptions (improving, e.g., the almost-isomorphism of Reinecke for m=2 to an actual isomorphism). The arguments rely on standard perfectoid toolkit (André’s lemma, arc covers, pullback squares from BS22 Cor. 8.12, p-root closure) and are therefore likely to be reusable. The work is a solid, incremental contribution to the explicit side of perfectoid algebra rather than a foundational breakthrough.

major comments (2)
  1. Theorem 2.11, last paragraph of the proof: the descent of “Cone killed by d after p-completely faithfully flat base change” to the original map is deferred entirely to “the first part of the proof of [Cai+25, Proposition 2.4.7]”. That reference is not self-contained for a reader who has not internalized Cai–Lee–Ma–Schwede–Tucker; a short explicit sketch (or a citation to a fully written general lemma on cones after p-completely faithfully flat base change) should be added so that the argument for the central monogenic claim stands on its own.
  2. Proposition 2.4 / Corollary 2.5 and the subsequent use in Remark 2.6 and Theorem 2.11: the nonzerodivisor conclusion for d on A_perfd rests on several lemmas from the authors’ concurrent preprint IN26 (Lemmas 2.3, 5.1, 5.2). While the logical dependence is legitimate, the manuscript should either reproduce the short statements needed or flag more clearly that the bounded-torsion hypothesis is used precisely to invoke those lemmas; otherwise the embedding A_perfd ⊆ A[1/d] looks more unconditional than it is.
minor comments (7)
  1. Title and abstract: “Perfectoidizaiton” is misspelled (should be “Perfectoidization”). The abstract also writes A_{pfd} while the body consistently uses A_perfd; unify the notation.
  2. Remark 2.10: the definition of the resultant via the determinant of the Sylvester-type map is correct, but the parenthetical claim that d = det(V)^2 “up to sign” should be made precise (including the usual leading-coefficient and sign factors) so that the equality “killed by d” in Theorem 2.11 is unambiguous.
  3. Example 2.12: the matrix of the trace form is displayed with an awkward line-break; a cleaner array would help. The observation that p A_perfd ⊆ A while A → A_perfd is not a p-almost isomorphism is useful and could be highlighted earlier as motivation for the discriminant control.
  4. Theorem 3.1 / Remark 3.3: the parenthetical remark that the discriminant t^{m−1} “aligns with Theorem 2.11” is true but terse; a one-sentence comparison would help the reader see the link.
  5. Notation 3.8 and 3.16: Fun_{d-acst} and Cont_{d-acst} are clear once defined, but an explicit sentence that these are subrings of the product / continuous functions under pointwise operations would avoid a momentary ambiguity.
  6. Several places (e.g., proof of Proposition 2.2, Claim inside it) use “■” and “□” inconsistently for end-of-proof markers; standardize.
  7. References: [IN26] and [Bha25] are preprints; update arXiv numbers / status if available at revision time. The Stacks Project tags are fine.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: monogenic discriminant bound and explicit Kummer/split formulae are derived from universal properties, resultant identities and base-change lemmas, not by redefining the target.

full rationale

The paper's central claims (Theorem 2.11 / 1.1: Cone(A o A_perfd) killed by the discriminant d, hence d A_perfd \subseteq A under the stated nonzerodivisor + bounded p^ heta-torsion hypotheses; density criterion Theorem 2.15; explicit descriptions of Kummer-type and split perfectoidizations in §3) are obtained by standard arguments: p-completely faithfully flat base change that splits m(t), evaluation map whose kernel/cokernel is killed by det(V) via Cramer's rule, descent of the cone-killing statement, and pullback squares from the universal property of perfectoidization (BS22 Cor. 8.12 / IN26 Prop. 2.7). Self-citations to the authors' prior work IN26 supply technical lemmas (discreteness of base change, injectivity after arc covers, p-root-closedness) that are used as tools; those lemmas do not already contain the monogenic discriminant bound or the explicit formulae of §3. No step equates a claimed prediction with a fitted input or with a definition of the target. The bounded-torsion hypothesis is stated explicitly and is not hidden. Score 1 reflects only the presence of non-load-bearing self-citation of prior technical lemmas.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The paper works entirely inside the standard axiomatic framework of commutative algebra and the Bhatt–Scholze theory of perfectoid rings and perfectoidization. No numerical free parameters appear. Load-bearing external inputs are existence/universality of A_perfd, arc-cover injectivity, and discreteness/base-change lemmas from BS22 and the authors’ IN26; these are domain assumptions of the subfield, not ad-hoc postulates invented here. No new physical or algebraic entities are introduced beyond notational rings (Fun_d-acst, Cont_d-acst) that are explicitly defined as subrings of products/function rings.

assumptions (5)
  • domain assumption Existence of an initial perfectoid A-algebra A_perfd for finite finitely presented A over perfectoid R (BS22 Thm 10.9), and that A → A_perfd is a d-isogeny when étale away from d.
    Used throughout as the definition of the object under study; not reproved.
  • domain assumption p-completely faithfully flat perfectoid covers and André’s lemma allowing monic polynomials to split after such base change (BS22 Thm 7.14).
    Invoked in the proof of Theorem 2.11 to reduce to the split Vandermonde case.
  • domain assumption Discreteness of derived p-complete base changes of finite free modules over bounded-torsion perfectoids, and injectivity of A_perfd into products along arc covers (IN26 Lemmas 5.1, 2.3, 5.2).
    Used in Prop 2.4, Cor 2.5, Thm 2.11, and pullback squares in §3.
  • standard math Standard commutative algebra: resultant/discriminant properties, Cramer’s rule for Vandermonde, Baire-category bounded torsion for derived p-complete modules (Stacks 0CQY), p-root closedness of perfectoids.
    Classical facts used for the monogenic bound and p-root-closure description (Prop 2.13).
  • domain assumption Bounded p^∞-torsion of R/dR (and often of A) so that d remains a nonzerodivisor on A_perfd.
    Explicit hypothesis of Thm 1.1 / 2.11 and Prop 2.4; without it the sharp containment is not claimed.
invented entities (1)
  • Fun_d-acst(n,R) / Cont_d-acst(X,R) independent evidence
    purpose: Explicit models for perfectoidizations of split finite algebras and of certain semiperfectoids as almost-constant (continuous) functions modulo (d)_perfd.
    Defined as concrete subrings of products/function rings; not new ontological objects, only convenient notation for the computed A_perfd.

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Pith. "Pith review of On Perfectoidizaiton of Finite Algebras over a Perfectoid Ring." pith.science (2026). https://pith.science/paper/I3VYAE7L

@misc{pith2026260612229,
  author       = {Pith},
  title        = {Pith review of: On Perfectoidizaiton of Finite Algebras over a Perfectoid Ring},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I3VYAE7L}},
  note         = {Machine review of arXiv:2606.12229}
}
abstract

We study general properties of the perfectoidization of finite algebras over a perfectoid ring, which helps to understand some precise and explicit descriptions. For example, we prove that if $A=R[t]/(m(t))$ where $m(t)$ is monic, $R$ is perfectoid and the discriminant $d$ of $m(t)$ is a non-zero divisor of $R$ satisfying a bounded torsion condition, then $dA_{\mathrm{pfd}}\subset A$. We also prove a density criterion reducing the construction of the perfectoidization to adjoining suitable $p$-power roots modulo $p$. In the second part of the paper, we compute perfectoidizations in several families of examples, including Kummer-type extensions and split finite algebras.

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