Relative Inverse Limit Perfection of Derived Commutative Rings

Daniel Fink

REVIEW 4 major objections 5 minor 24 references

Relative Inverse Limit Perfection of Derived Commutative Rings

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Relative perfectness coincides with formal étaleness for Noetherian F-finite rings in characteristic p, and every F-finite map factors through a relative inverse limit perfection.

desk verdict A substantial new construction with a real black-box problem: the Noetherian conclusions depend on Gabber's Remark 13.6, quoted without statement, and on a base-change step in Proposition 4.18 that needs a real proof. read the letter →

arxiv 2506.10626 v1 pith:YBXTAWKC submitted 2025-06-12 math.AC

Daniel Fink This is my paper · ORCID

classification math.AC MSC 13A3513B4013D03

keywords relativeFrobeniusF-finitenessrelativelyperfectalgebrasinverselimitperfectionanimatedringsderivedcommutativecotangentcomplexformalétaleness

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

Over a field of characteristic $p$, the Frobenius map has a relative version for a map $R \to S$, and the paper's goal is to understand how far a general map is from being relatively perfect. It constructs the relative inverse limit perfection $S^{\mathrm{perf}/R}$ as the inverse limit of the relative Frobenius tower, and shows that under an F-dualizability condition on $R$ this is a right adjoint to the inclusion of relatively perfect $R$-algebras. With this tool, it proves that every map of F-finite animated (connective derived) $\mathbb{F}_p$-algebras factors as a free finite-type map, then a relatively perfect map, then a surjection on $H^0$; for Noetherian input the middle term is Noetherian, and in the discrete Noetherian case it is regular. It also proves that for maps of Noetherian F-finite $\mathbb{F}_p$-algebras, relative perfectness is exactly formal étaleness, equivalently the vanishing of the cotangent complex. A reader should care because this gives a uniform finite-generation-style decomposition and a Frobenius-theoretic characterization of étale maps in positive characteristic.

What carries the argument

The load-bearing construction is the relative Frobenius tower of $S$ over $R$: the inverse system $\cdots \to S \otimes^{\mathrm{L}}_{R,F^3} R \to S \otimes^{\mathrm{L}}_{R,F^2} R \to S \otimes^{\mathrm{L}}_{R,F} R \to S$, where $F$ is the absolute Frobenius and the transition maps are relative Frobenius maps. Its inverse limit $S^{\mathrm{perf}/R}$ is the relative inverse limit perfection. The key identity is that if $F_*R$ is a dualizable $R$-module, then the functor $(-) \otimes^{\mathrm{L}}_{R,F} R$ commutes with limits, so the limit $S^{\mathrm{perf}/R}$ is itself relatively perfect; Lemma 2.9 supplies the companion identity $L_{F_{S/R}} \simeq L_{S/R} \oplus (L_{S/R} \otimes^{\mathrm{L}}_{S,F} S[1])$, which immediately gives vanishing of $L_{S/R}$ for relatively perfect maps. The category of relatively perfect $R$-algebras is presentable and closed under the needed limits, so the inclusion admits the relative inverse limit perfection as a right adjoint. For discrete rings, relative perfectness also requires Tor-independence of $S$ and $F_*R$ over $R$; in the derived setting the equivalence of the derived relative Frobenius map is enough.

What would settle it

Exhibit a map $R \to S$ of Noetherian F-finite $\mathbb{F}_p$-algebras with vanishing cotangent complex $L_{S/R}$ whose relative Frobenius $F_{S/R}: S \otimes_{R,F} R \to S$ is not an isomorphism; Theorem E says no such map exists. Equivalently, compute the relative inverse limit perfection $T = S^{\mathrm{perf}/R}$ for a relatively semiperfect map $R = \mathbb{F}_p[x_1,\dots,x_n] \to S$ with $S$ Noetherian and check whether $T$ is regular Noetherian, since a single failure would falsify Proposition 4.18 and the Noetherian part of Theorem 5.7.

Watch

Extended reading notes

Core claim

The paper's central claim is that relative perfectness is the right measure of Frobenius-invariant structure in derived positive-characteristic algebra, and that every map of F-finite animated $\mathbb{F}_p$-algebras can be built from one free finite-type step, one relatively perfect step, and one surjection. Concretely, Theorem 5.7 factors $R \to S$ as $R \to R[x_1,\dots,x_n] \to T \to S$ with the first map the obvious free extension, the second relatively perfect (relative Frobenius an equivalence), and the third surjective on $H^0$; if $H^0(R)$ is Noetherian, $T$ is Noetherian as well, and the construction controls coconnectivity in the derived setting. In the discrete Noetherian case Corollary 5.8 makes $T$ discrete and regular when both rings are Noetherian. The companion Theorem 5.3 states that a map $R \to S$ of Noetherian F-finite $\mathbb{F}_p$-algebras is relatively perfect if and only if the cotangent complex $L_{S/R}$ vanishes, which for these rings is the same as formal étaleness. This equivalence turns the factorization into a geometric statement: formally smooth morphisms of locally Noetherian F-finite $\mathbb{F}_p$-schemes factor locally as a projection from affine $n$-space followed by a formally étale morphism.

Load-bearing premise

The Noetherian and regularity conclusions rest on an external black box, the remark cited as [7, Remark 13.6], which asserts that the inverse limit perfection over a polynomial ring of a Noetherian F-finite ring is regular Noetherian; if that assertion requires extra hypotheses, the regularity and Noetherian statements in the main theorems would need revision, although the existence of the factorization itself may survive.

Editorial extensions

If this is right

Every F-finite animated $\mathbb{F}_p$-algebra $S$ admits a polynomial $\mathbb{F}_p$-algebra $R'$ and a relatively perfect $R'$-algebra $T$ with $T \to S$ surjective on $H^0$; when $H^0(S)$ is Noetherian, $T$ is regular Noetherian.
A map of Noetherian F-finite $\mathbb{F}_p$-algebras with acyclic cotangent complex is relatively perfect, so the $I$-adic completion of a Noetherian F-finite ring is a relatively perfect algebra over it.
Formally smooth morphisms of locally Noetherian F-finite $\mathbb{F}_p$-schemes decompose, Zariski locally, as a projection from affine $n$-space followed by a formally étale morphism.
The discrete factorization gives an L-smooth-by-surjective factorization in the sense of earlier work, reproducing and refining that statement by an explicit construction.
Relative perfectness, rather than plain perfectness, is the Frobenius-theoretic property that matches formal étaleness and vanishing cotangent complex in the Noetherian F-finite setting.

Reading between the lines

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

The realization of derived completion as a relative Frobenius tower limit suggests that relative inverse limit perfection is a completion operation interpolating between $p$-adic completion and absolute perfection; the paper exhibits this for polynomial bases, but its scope over general bases is left implicit.
The right-adjoint formulation may allow a relative perfection operation for arbitrary maps of schemes, giving a Frobenius-theoretic closure whose fixed points are exactly the maps that are étale in the appropriate derived sense.
Theorem E's equivalence between relative perfectness and formal étaleness may extend to non-Noetherian F-finite rings if the Tor-independence condition in the definition is replaced by a derived version; the paper establishes the Noetherian case only.
The same machinery could be tested as a tool for constructing p-bases or proving regularity criteria in settings where the Noetherian hypothesis is dropped, since the inverse limit perfection is defined without assuming Noetherianity.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a genuine contribution, and you should send it to referees. But the Noetherian conclusions are not as solid as the formal parts, and the referee should insist on closing a real gap.

The new material is real. The relative inverse limit perfection functor, the adjunction relative to relatively perfect algebras, the free-by-relatively-perfect-by-surjective factorization for F-finite animated rings (Theorem 5.7), and the converse to cotangent-complex vanishing for Noetherian F-finite rings (Theorem 5.3) are all new and natural. The formal framework is coherent, and the proofs I checked are mostly standard. I do not share the reader's worry about Lemma 2.9: the 'straightforward calculation' is fine, since d(r^p)=0 in characteristic p.

The soft spot is Proposition 4.18, and the stress-test note is right to put its finger there. The paper invokes Gabber's Remark 13.6 without stating the hypotheses or the content of the result. That might be acceptable in a research article if the application were straightforward, but here the remark is load-bearing for the Noetherianness of the relative inverse limit perfection. Worse, the same proposition asserts S^{perf/A} ⊗_{R^{perf/A}} R ≃ S^{perf/R} in one sentence, justified by saying that R is a perfect complex over the regular ring R^{perf/A} and hence base change preserves limits. That step is not automatic: a quotient of a regular ring need not be perfect as a module over that regular ring. The base-change comparison needs an argument, or an extra hypothesis. If it fails, the Noetherian and regularity conclusions in Lemma 5.2(ii), Theorem 5.7(ii), Corollary 5.8, and the proof of Theorem 5.3 do not go through. The existence part of the factorization would likely survive, since it does not need the Noetherian conclusions.

To be clear, I did not find an internal contradiction in the remaining formal parts. The reliance on the unpublished manuscript [3] is a smaller issue; those references are mostly for Adams completion facts, and quoting the precise statements would make the paper more checkable.

The paper is for specialists in F-singularities and derived commutative algebra. It deserves a serious referee. The referee should ask the author to either prove the needed version of Gabber's remark, or quote it exactly, and to give a complete proof of the base-change identity in Proposition 4.18. My recommendation: accept for peer review, with the expectation of a revision.

Referee Report

4 major / 5 minor

Summary. The paper introduces relative analogues of Frobenius finiteness, semiperfectness, and perfectness for maps of derived commutative F_p-algebras, constructs a relative Frobenius tower and its inverse limit perfection, and proves that this perfection is a right adjoint to the inclusion of relatively perfect algebras when the base is F-dualizable. In the animated setting it proves a factorization theorem for F-finite maps into a free finite-type map, a relatively perfect map, and a surjection on H0, with coconnectivity and Noetherian conclusions, and uses this to prove a converse statement relating vanishing of the cotangent complex to relative perfectness for Noetherian F-finite rings, yielding a formal étaleness characterization.

Significance. The construction is natural and the main factorization theorem is a useful structural result in positive characteristic. The paper is well organized and largely explicit, with clean adjunction statements, Tor-independence results, and applications such as Corollaries C through E and the factorization of formally smooth morphisms. Its strongest advertised consequences, however, currently depend on Gabber's unpublished Remark 13.6 and the unpublished manuscript [3], as well as on several sketched arguments; the Noetherian and regularity conclusions should be considered conditional until those inputs are stated and proved.

major comments (4)

[Proposition 4.18] The proof that S^{perf/R} is regular Noetherian has two gaps. First, it does not state the hypotheses or content of Gabber's Remark 13.6 beyond the citation in Example 4.17. Second, the displayed chain of equivalences, particularly S^{perf/A} \otimes_{R^{perf/A}} R \simeq S^{perf/R}, is asserted in one sentence using preservation of limits; base change does not commute with inverse limit perfection merely because the base is a perfect complex, and the tower over R^{perf/A} must be shown to base-change to the tower over R. Since Lemma 5.2(ii), Theorem 5.7(ii), Corollary 5.8, and the proof of Theorem 5.3 all use this proposition, these claims are not yet established.
[Lemma 2.9] The main computation is omitted. The proof reduces to F_p[X] -> F_p[X,Y] and says the vanishing follows from a "straightforward calculation", but this vanishing is the key input for Proposition 3.11 and hence for the acyclicity of cotangent complexes of relatively perfect maps. Please write out the calculation and justify the passage from the polynomial case to a general animated ring through the sifted colimit; cotangent complexes do not in general commute with arbitrary colimits, so the relevant preservation statement needs to be cited or proved.
[Theorem 5.3] The proof of Theorem 5.3 invokes Lemma 5.2 to obtain a Noetherian T, and then uses the implication "étale implies relatively perfect". This implication is asserted in Example 3.6 but not proved for the relative notion; the cited references support étale implies weakly étale and related statements, but the paper does not give the direct verification that the relative Frobenius map is an equivalence. Since Theorem 5.3 is the converse direction behind Theorem E, this step should be proved directly for Noetherian F-finite rings or supported by a precise reference proving exactly this implication.
[Propositions 4.6, 4.8, 4.19] Several auxiliary results depend on the unpublished manuscript [3]: Proposition 4.6, Proposition 4.8, and Proposition 4.19 quote specific numbered results from [3] without reproducing their statements. This is acceptable for a preprint, but for a journal submission the dependence should be made self-contained, or the statements should be quoted in enough detail for the reader to verify the hypotheses.

minor comments (5)

[Lemma 2.9] The word "Frobenus" should be "Frobenius".
[Lemma 3.8] In part (i), "an derived" should read "a derived".
[Example 4.17] In the displayed square, the upper horizontal map is described as "the Frobenius on R", but the rings displayed are polynomial rings over S; the description should be rephrased to identify the map precisely.
[Corollary 5.4] The phrase "the discrete relative inverse limit perfection" should be reconciled with Definition 3.1, since the relative inverse limit perfection was defined for derived rings; please clarify the precise object being claimed relatively perfect.
[Remark 2.2 and Lemma 2.9] The same sifted colimit presentation is cited to two different locations in [20]; the references should be unified.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: main factorization and cotangent-complex results are derived from explicit definitions and external references; only auxiliary self-citations to the companion preprint [5] appear.

full rationale

The core derivation chain is self-contained. Relative perfectness, relative semiperfectness, and the relative Frobenius tower are defined explicitly (Definitions 3.1, 4.1 and Construction 3.20), and the relative inverse limit perfection is constructed and proved to be relatively perfect and right-adjoint without assuming the target theorems (Lemmas 3.25, 3.26, Proposition 4.4). The factorization results in Lemma 4.15, Construction 5.1, Lemma 5.2, and Theorem 5.7 are obtained directly from these constructions together with standard external facts, not by quoting the conclusions. Theorem 5.3 and Theorem E follow from Lemma 4.14, Proposition 3.11, Lemma 5.2, and Stacks-project/EGA references [8,23]; they do not reduce to their own statements. The only self-citations are to the companion preprint [5]: Proposition 2.15 is used in Corollary 5.6 and Proposition 3.11 is used in Corollary 5.4, and Corollary 5.8 is described as giving a new proof of [5, Theorem 4.4]. These citations support auxiliary corollaries, not the paper's central claims, so they are not load-bearing circularity. The Noetherian and regularity conclusions in Proposition 4.18, Lemma 5.2(ii), and Theorem 5.7(ii) rely on Gabber's Remark 13.6, quoted only indirectly through Example 4.17; this is an external black-box dependence and a verification concern, not a circular reduction. Similarly, the one-sentence base-change identification Sperf/A ⊗_{Rperf/A} R ≃ Sperf/R in Proposition 4.18 is not expanded and may require hypotheses, but it is not an equation identifying a prediction with its input. No circular step can be exhibited.

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

The central claims rest on standard derived algebraic geometry background and on several black-box results, most notably Gabber's inverse limit perfection over polynomial rings and Adams completion facts from an unpublished manuscript. No numerical parameters are fitted to data, and no new entities are postulated.

assumptions (5)

domain assumption Absolute Frobenius functor on derived F_p-algebras exists and preserves limits and colimits.
Invoked in Section 2 via [10, Construction 2.4.1]; it underlies the definition of the relative Frobenius map and the behavior of the Frobenius tower.
standard math Absolute Frobenius induces the zero map on negative cohomology groups of derived F_p-algebras.
Used in Lemma 2.6 and Lemma 2.7, attributed to [4, Proposition 11.6]. It is essential for proving coconnectivity of the inverse limit perfection.
domain assumption Gabber's Remark 13.6: the inverse limit perfection over a polynomial ring of a Noetherian F-finite ring is regular Noetherian.
Load-bearing for the Noetherian and regularity assertions in Proposition 4.18, Lemma 5.2, and Theorem 5.7. The paper cites [7, Remark 13.6] without proof.
domain assumption Adams completion agrees with derived I-adic completion for maps R -> S with surjective H0 and finitely generated kernel.
Used in Propositions 4.6 and 4.8, cited from the unpublished manuscript [3, Proposition 3.2.5]. This is a black box that the reader cannot independently verify from the provided text.
standard math For a map of Noetherian F-finite rings, vanishing of the cotangent complex implies formal etaleness, and formally etale maps are relatively perfect.
Used in Theorem 5.3 via [8, Theoreme 21.2.7] and [23, Lemma 0EBS]. These are classical or Stacks-project results.

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Cite this review

Pith. "Pith review of Relative Inverse Limit Perfection of Derived Commutative Rings." pith.science (2026). https://pith.science/paper/YBXTAWKC

@misc{pith2026250610626,
  author       = {Pith},
  title        = {Pith review of: Relative Inverse Limit Perfection of Derived Commutative Rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YBXTAWKC}},
  note         = {Machine review of arXiv:2506.10626}
}

read the original abstract

We study the relative Frobenius map associated with a map of derived commutative rings over a field of positive characteristic. As part of this, we examine a relative analog of perfectness and construct a relative inverse limit perfection which, under suitable conditions on the base, serves as a right adjoint to the inclusion of relatively perfect algebras into the category of all algebras. Specializing to animated rings, we investigate relative versions of semiperfectness and F-finiteness, and use these to show that any map of F-finite animated rings factors into a free map of finite type, followed by a relatively perfect map, followed by a surjective map. We also show that, for a morphism of Noetherian F-finite rings, the vanishing of the cotangent complex implies that the morphism is relatively perfect.

Discussion (0). Continue with ORCID to comment.

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Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [3]

F-finite schemes have a dualizing complex

Bhargav Bhatt, Manuel Blickle, Karl Schwede, and Kevin Tucker. “F-finite schemes have a dualizing complex”. Unpublished article. 28.10.2024

work page 2024

[2] [1]

Homologie des algèbres commutatives

Michel André. Homologie des algèbres commutatives . French. V ol. 206. Grundlehren Math. Wiss. Springer, Cham, 1974

work page 1974

[3] [2]

p-adic derived de Rham cohomology

Bhargav Bhatt. p-adic derived de Rham cohomology. 2012. arXiv: 1204.6560 [math.AG] . URL: https: //arxiv.org/abs/1204.6560

arXiv 2012

[4] [4]

Projectivity of the Witt vector affine Grassmannian

Bhargav Bhatt and Peter Scholze. “Projectivity of the Witt vector affine Grassmannian”. English. In: Invent. Math. 209.2 (2017), pp. 329–423. ISSN : 0020-9910. DOI: 10.1007/s00222-016-0710-4

doi:10.1007/s00222-016-0710-4 2017

[5] [5]

$L$-smooth factorization for Noetherian $F$-finite rings

Manuel Blickle and Daniel Fink. L-smooth factorization for Noetherian F -finite rings. 2025. arXiv: 2501 . 09437 [math.AC]. URL: https://arxiv.org/abs/2501.09437

work page Pith review arXiv 2025

[6] [6]

Purity for flat cohomology

K˛ estutis ˇCesnaviˇcius and Peter Scholze. “Purity for flat cohomology”. English. In:Ann. Math. (2) 199.1 (2024), pp. 51–180. ISSN : 0003-486X. DOI: 10.4007/annals.2024.199.1.2

doi:10.4007/annals.2024.199.1.2 2024

[7] [7]

Notes on some t-structures

Ofer Gabber. “Notes on some t-structures”. English. In: Geometric aspects of Dwork theory. Vol. I, II . Berlin: Walter de Gruyter, 2004, pp. 711–734. ISBN : 3-11-017478-2

work page 2004

[8] [8]

Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de sché- mas. (Première partie). Rédigé avec la colloboration de J. Dieudonné

A. Grothendieck. “Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de sché- mas. (Première partie). Rédigé avec la colloboration de J. Dieudonné”. French. In: Publ. Math., Inst. Hautes Étud. Sci. 20 (1964), pp. 101–355. ISSN : 0073-8301. DOI: 10.1007/BF02684747

work page doi:10.1007/bf02684747 1964