{"id":"30d8f8ec-d91e-405b-8092-3e346d5b3c17","arxiv_id":"2606.12229","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For monogenic finite algebras over a perfectoid ring with nonzerodivisor discriminant d of bounded p-torsion, d A_perfd sits inside A, and perfectoidizations of Kummer and split extensions are computed explicitly.","lead":"The paper proves that perfectoidizations of finite monogenic algebras over perfectoid rings stay controlled by the discriminant: d times the perfectoidization sits inside the original algebra. It also gives explicit computations for Kummer and split extensions, turning an abstract existence theorem into concrete rings.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader's identification of the bounded-torsion hypothesis as the weakest load-bearing assumption is accurate and already reflected in the paper's own statements (Thm 2.11, Rem 2.6, Prop 2.4). The proof of the monogenic bound itself is short, uses only standard tools (André's lemma / p-completely flat splitting, Vandermonde/Cramer, arc-cover injectivity of perfectoidization), and correctly descends. The density criterion (Thm 2.15) and the explicit Kummer/split computations (Thms 3.1, 3.6, 3.9) are independent and carefully checked. No further load-bearing concern surfaces under good-faith scrutiny; the ACCEPT verdict with low correctness risk stands.","tokens_in":19422,"tokens_out":575,"duration_ms":6063,"concrete_test":"Take a concrete perfectoid R (e.g. the p-adic completion of Z_p[p^{1/p^\theta}]) and a monic m(t) whose discriminant d is a nonzerodivisor with R/dR of unbounded p^\theta-torsion (if such an example is known to exist); compute A_perfd directly via the prismatic or almost-purity description and check whether d A_perfd still lands inside A. If the containment fails, the hypothesis is sharp; if it holds, the hypothesis can be weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central monogenic claim (Theorem 2.11 / 1.1) is proved by a standard arc-cover + Cramer argument after a p-completely faithfully flat base change that splits m(t): the evaluation map A' \to (R')^n has kernel/cokernel killed by det(V), hence Cone(A' \to A'_perfd) is killed by d = det(V)^2 (up to units/sign), and the claim descends by the first half of the argument of Cai+25 Prop. 2.4.7. The bounded p^\theta-torsion hypothesis on R/dR is used only to guarantee that d remains a nonzerodivisor on A_perfd (via Prop. 2.4 / Cor. 2.5 and the arc-cover injectivity of IN26 Lem. 2.3), so that the embedding A_perfd \to A[1/d] is honest rather than almost. This is an explicit, standard hypothesis already flagged by the reader; it is not hidden, and the paper correctly states the conclusion under it. No internal gap, circularity, or unstated assumption appears in the argument.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","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).","tokens_in":19706,"tokens_out":1363,"duration_ms":23911,"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":[{"comment":"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.","section":null},{"comment":"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.","section":null}],"minor_comments":[{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"Several places (e.g., proof of Proposition 2.2, Claim inside it) use “■” and “□” inconsistently for end-of-proof markers; standardize.","section":null},{"comment":"References: [IN26] and [Bha25] are preprints; update arXiv numbers / status if available at revision time. The Stacks Project tags are fine.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is technically solid and the central monogenic statement is correctly proved under the stated hypotheses; the only real presentation debt is the opaque descent citation to Cai+25 and the heavy reliance on the authors’ own IN26 lemmas. I see no novelty or priority issue. Fit for a solid algebra journal is good; the contribution is incremental but useful. Title typo should be fixed before any public version."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new content is real and usable. BS22 already gives existence of A_perfd and that A \to A_perfd is a d-isogeny. What this paper adds is the monogenic sharpening (Thm 2.11): Cone(A \to A_perfd) is killed by the discriminant d itself, so under the usual nonzerodivisor + bounded p^\\infty-torsion hypotheses one gets the honest containment d A_perfd \\subseteq A and the embedding into A[1/d]. That improves the general J^{n0} statement from Cai+25. The density criterion (Thm 2.15 / Cor 2.16) is clean: if you have an intermediate C with C/pC semiperfect, its p-completion is already A_perfd when V(d)=V(p). Then Section 3 delivers the explicit models people actually need—Kummer R[t]/(t^m - r) becomes the completed p-power roots of the m-th root (Thms 3.1, 3.6), and split monics become the “almost constant” function rings Fun_{d-acst}(n,R) (Thm 3.9). The Hopf-algebra and semiperfectoid applications are nice bonuses.\n\nProofs are standard perfectoid toolkit done carefully: André splitting + Vandermonde/Cramer for the discriminant, arc-cover injectivity from their earlier IN26 lemmas, pullback squares from BS22 Cor 8.12, p-root closure. No circular redefinition; the self-citations to IN26 are prior technical lemmas, not load-bearing inventions. The bounded-torsion hypothesis is stated up front and is exactly what turns “almost” into “honest”; without it the statements correctly become almost. Title has a typo (“Perfectoidizaiton”), but that is cosmetic.\n\nThis is for people who compute with perfectoidizations of finite algebras, almost purity, or arc covers. It will be cited by that group. The math is solid, the examples are concrete, and it deserves a serious referee. I would accept for peer review and would cite the monogenic bound and the Kummer formulas myself.","headline":"Solid specialist paper: sharp monogenic discriminant bound d A_perfd ⊆ A plus explicit Kummer/split models that people will actually use.","tokens_in":20353,"tokens_out":545,"would_cite":true,"duration_ms":24851,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14G45","13J10"],"pacs":[],"model":"grok-4.5","headline":"Perfectoidization of monic finite algebras over a perfectoid ring is controlled by the discriminant: d times the perfectoidization sits inside the original algebra.","keywords":["perfectoid rings","perfectoidization","discriminant","Kummer extensions","bounded torsion","p-root closure","finite algebras","almost purity"],"falsifier":"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.","tokens_in":20326,"feed_emoji":"📐","tokens_out":910,"duration_ms":7682,"temperature":0.7,"pith_summary":"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.","feed_headline":"Discriminant controls perfectoidization of finite algebras","feed_subtitle":"d times the perfectoidization sits inside the original monogenic algebra over a perfectoid ring","key_machinery":"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).","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Discriminant d forces d A_perfd inside monogenic A over perfectoid R","Cone of monogenic A to its perfectoidization killed by disc(m)","Bounded p-torsion makes A_perfd d-torsion-free and inside A[1/d]","Density reduces perfectoidization to adjoining p-power roots mod p","Explicit perfectoidizations computed for Kummer and split finite algebras"],"cache_read_input_tokens":10624,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Discriminant d forces d A_perfd inside monogenic A over perfectoid R","Cone of monogenic A to its perfectoidization killed by disc(m)","Bounded p-torsion makes A_perfd d-torsion-free and inside A[1/d]","Density reduces perfectoidization to adjoining p-power roots mod p","Explicit perfectoidizations computed for Kummer and split finite algebras"]},"model":"grok-4.5","effort":"low","cost_usd":0.007344,"raw_usage":{"total_tokens":1750,"prompt_tokens":693,"num_sources_used":0,"completion_tokens":106,"cost_in_usd_ticks":73440000,"prompt_tokens_details":{"text_tokens":693,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":951,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":693,"tokens_out":106,"duration_ms":8289,"temperature":1.0,"reasoning_tokens":951,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T14:16:52.535868+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"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.","supporting_citations":[],"review_version":2}