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The crystalline comparison of Ainf-cohomology: the case of good reduction

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A canonical Frobenius-compatible map links crystalline cohomology to A_inf-cohomology for formal schemes with good reduction.

desk verdict A clearly written re-proof of known BMS comparison theorems via a new functorial crystalline map h_crys; the load-bearing local lemmas are compressed, but the paper deserves refereeing. read the letter →

arxiv 1908.06366 v1 pith:IO3NPDJB submitted 2019-08-18 math.AG math.NT

Zijian Yao This is my paper · ORCID
classification math.AGmath.NT MSC 14F3014F4014G22
keywords A_inf-cohomologycrystallinecohomologyintegralp-adicHodgetheoryquasiregularsemiperfectoidringsquasisyntomicdescentBreuil-Kisin-FarguesmodulesB_criscomparisongoodreduction
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

Integral p-adic Hodge theory studies the $A_{\mathrm{inf}}$-cohomology $\mathrm{R}\Gamma_{A_{\mathrm{inf}}}(X)$ of a smooth formal scheme over $O_C$, where $A_{\mathrm{inf}}=W(O_C^\flat)$ is the period ring built from the tilt of $O_C$ and $A_{\mathrm{cris}}$ is its divided-power envelope. This paper explains that object crystallinely: it constructs a canonical Frobenius-equivariant map from the crystalline cohomology of $X_{O_C}/p$ over $A_{\mathrm{cris}}$ into $\mathrm{R}\Gamma_{A_{\mathrm{inf}}}(X)\otimes^{\mathbf L}_{A_{\mathrm{inf}}}A_{\mathrm{cris}}$. The map is assembled locally on quasiregular semiperfectoid covers, ring-theoretic analogues of local complete intersections, where the relevant $A_{\mathrm{cris}}$-algebra is topologically free and the projection to $S/p$ carries divided powers. The result matters because the existence of this single map, in the direction from crystalline to $A_{\mathrm{inf}}$-cohomology, is enough to reprove the principal comparison theorems of the area: crystalline specialization, the $B_{\mathrm{cris}}$ comparison, the Breuil-Kisin-Fargues structure of the cohomology groups, and compatibility with Hodge filtrations, without the heavier relative de Rham-Witt and all-coordinates machinery.

What carries the argument

The load-bearing local object is the class of quasiregular semiperfectoid $O_C$-algebras $S$: p-torsion-free quotients of perfectoid rings by quasiregular ideals, with examples such as $O_C\langle X^{1/p^\infty}\rangle/(X)$. For such $S$, Lemmas 2.6 and 2.7 show that $A\Omega_S\otimes^{\mathbf L}A_{\mathrm{cris}}$ is a topologically free $A_{\mathrm{cris}}$-module concentrated in degree $0$, and that the natural projection $\beta:A\Omega_S\otimes^{\mathbf L}A_{\mathrm{cris}}\to S/p$ is a PD-thickening, meaning its kernel carries divided powers. This makes $A\Omega_S\otimes^{\mathbf L}A_{\mathrm{cris}}$ an object of the crystalline site of $R/p$ over $(A_{\mathrm{cris}},O_C/p,\gamma)$, so restriction along $R\to S$ produces a canonical map from $\mathrm{R}\Gamma_{\mathrm{crys}}((R/p)/A_{\mathrm{cris}})$ to $A\Omega_S\otimes^{\mathbf L}A_{\mathrm{cris}}$; quasisyntomic descent over such $S$, followed by homotopy limits over affine opens of $X_{\mathrm{et}}$, assembles these local maps into the global $h_{\mathrm{crys}}$.

What would settle it

Compute the divided-power divisibility directly on the explicit ring $S_0=O_C\langle X_j^{1/p^\infty}\rangle_{j\in J}/(X_j)$: take $x$ in the kernel of $A\Omega_{S_0}\otimes^{\mathbf L}A_{\mathrm{cris}}\to S_0$ corresponding to $X_j$ and check whether $x^p$ lies in $p$ times that kernel. A single $j$ where the divisibility fails falsifies Lemma 2.7 and with it the local construction of Section 3.1.

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Extended reading notes

Core claim

Theorem 1 of the paper asserts that for a smooth formal scheme $X$ over $\mathrm{Spf}\,O_C$ there is a functorial $\varphi$-equivariant map $h_{\mathrm{crys}}:\mathrm{R}\Gamma_{\mathrm{crys}}((X_{O_C}/p)/A_{\mathrm{cris}})\to \mathrm{R}\Gamma_{A_{\mathrm{inf}}}(X)\otimes^{\mathbf L}_{A_{\mathrm{inf}}}A_{\mathrm{cris}}$ that becomes the inverse of the de Rham comparison $\gamma_{\mathrm{dR}}$ after base change to $O_C/p$. The direction is opposite to the map previously obtained by taking limits over coordinate choices, and the paper shows that existence alone is sufficient: derived Nakayama's lemma yields the crystalline specialization $\mathrm{R}\Gamma_{A_{\mathrm{inf}}}(X)\otimes^{\mathbf L}_{A_{\mathrm{inf}},\vartheta}W(k)\simeq \mathrm{R}\Gamma_{\mathrm{crys}}(X_k/W(k))$; after base change to $B_{\mathrm{cris}}$ and composition with the etale comparison one obtains the $B_{\mathrm{cris}}$ comparison isomorphism; and a descending induction on the cohomological degree, using the equality of $W(k)$-rank and $\mathbb Z_p$-rank, proves that each $\mathrm{H}^i_{A_{\mathrm{inf}}}(X)$ is a Breuil-Kisin-Fargues module (a finitely presented $A_{\mathrm{inf}}$-module whose Frobenius becomes an isomorphism after inverting $\xi$ and which is free after inverting $p$). A $B_{\mathrm{dR}}^+$-version, formulated through an infinitesimal site on the generic fibre, supplies the filtration compatibility of the comparison.

Load-bearing premise

The load-bearing premise is the local lemma that for every p-torsion-free quasiregular semiperfectoid $O_C$-algebra $S$, the projection $A\Omega_S\otimes^{\mathbf L}A_{\mathrm{cris}}\to S/p$ is a divided-power thickening; if that divisibility statement fails, the map $h_{\mathrm{crys}}$ cannot be assembled.

Editorial extensions

If this is right

  • The crystalline comparison $\mathrm{R}\Gamma_{A_{\mathrm{inf}}}(X)\otimes^{\mathbf L}_{A_{\mathrm{inf}},\vartheta}W(k)\simeq\mathrm{R}\Gamma_{\mathrm{crys}}(X_k/W(k))$ follows by derived Nakayama's lemma once $h_{\mathrm{crys}}$ exists.
  • The $B_{\mathrm{cris}}$ comparison gives a $(\mathrm{Gal}_K,\varphi)$-equivariant isomorphism $\mathrm{H}^i_{\mathrm{crys}}(X_{k_0}/W(k_0))\otimes B_{\mathrm{cris}}\simeq\mathrm{H}^i_{\mathrm{et}}(X_C^{\mathrm{ad}},\mathbb Z_p)\otimes B_{\mathrm{cris}}$, so $\mathrm{H}^i_{\mathrm{et}}(X,\mathbb Q_p)$ is a crystalline Galois representation.
  • Each group $\mathrm{H}^i_{A_{\mathrm{inf}}}(X)$ is a Breuil-Kisin-Fargues module, and the cohomology vanishes for $i>2\dim X/O_C$.
  • The torsion inequalities $\mathrm{length}_{W(k)}(\mathrm{H}^i_{\mathrm{crys}}(X_k/W(k))_{\mathrm{tor}}/p^n)\ge\mathrm{length}_{\mathbb Z_p}(\mathrm{H}^i_{\mathrm{et}}(X,\mathbb Z_p)_{\mathrm{tor}}/p^n)$ hold for every $n\ge 1$.
  • When $\mathrm{H}^i_{\mathrm{crys}}(X_k/W(k))$ and $\mathrm{H}^{i+1}_{\mathrm{crys}}(X_k/W(k))$ are torsion free, the integral crystalline cohomology with its Frobenius action is recovered from the etale cohomology together with the $B_{\mathrm{dR}}^+$-lattice $\mathrm{H}^i_{\mathrm{inf}}(X/B_{\mathrm{dR}}^+)$.

Reading between the lines

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

  • Because the global map is assembled purely from local divided-power data, the whole comparison can be audited object-by-object: verifying the PD-thickening statement on the explicit rings $O_C\langle X_j^{1/p^\infty}\rangle/(X_j)$ would certify the engine of the proof independently of the surrounding descent formalism.
  • Remark 3.8 leaves open whether $h_{\mathrm{crys}}$ is itself a quasi-isomorphism; if it were, the missing $A_{\mathrm{cris}}$-specialization statement would follow, but the non-finite-generation of $\ker(A_{\mathrm{cris}}\to O_C/p)$ blocks the obvious derived Nakayama argument.
  • The author suggests the construction adapts to logarithmic settings; if the local PD-thickening lemma survives that variant, the same quasisyntomic-descent assembly would plausibly carry the comparison theorems to semistable reduction without a new global analysis.
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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

3 major / 5 minor

Summary. The paper proposes a new construction of the crystalline comparison for A_inf-cohomology of smooth formal schemes over O_C in the case of good reduction. The main object is a functorial phi-equivariant map h_crys: RGamma_crys((X_O_C/p)/A_cris) -> RGamma_Ainf(X) tensor^L_{A_inf} A_cris, compatible with the de Rham comparison after base change (Theorem 1 and Theorem 3.6). The proof reduces via quasisyntomic descent to quasiregular semiperfectoid algebras S, for which it must show that AOmega_S tensor^L A_cris is a discrete topologically free A_cris-algebra (Lemma 2.6) and that its projection to S/p is a PD-thickening (Lemma 2.7). From this map the paper derives the crystalline specialization to W(k), the Breuil-Kisin-Fargues structure on H^i_Ainf(X), the B_cris comparison for Galois representations, and, via a new infinitesimal B_dR^+ site in Section 5, the filtration compatibility and some integral recovery results. The paper explicitly takes the de Rham comparison as an input and therefore does not aim to reprove that half of [BMS18].

Significance. If the local lemmas are correct, the paper gives a genuinely simpler route to the main comparison theorems of [BMS18], avoids relative de Rham-Witt machinery, and packages the B_dR^+ side in a way adapted to filtration statements. The direction of h_crys is well chosen for Galois-invariant arguments, and the derivation of Corollaries 3.7 and 4.7 and of Theorem 4.13 from a single functorial map is clean. The paper is transparent about its inputs and about the omitted local computation in Remark 2.8; there is no fitting and no circularity, since the target comparison is not used to prove itself. The significance is contingent on completing the proofs of Lemmas 2.6 and 2.7, which are the load-bearing local statements.

major comments (3)
  1. [§2.4, Lemma 2.6] The proof is too abbreviated for a load-bearing statement. It asserts that LOmega(S/p) is a free O_C/p-module concentrated in degree 0 by 'considering the graded pieces wedge^i L(S/p)[-i] for the conjugate filtration', and that the D(A_inf)-valued sheaf R |-> AOmega_R 'takes discrete values on quasi-regular semiperfectoid objects'. The second assertion is essentially the discreteness half of what has to be proved, and the first does not by itself give a basis of the derived (p,xi)-completion AOmega_S. One needs an argument that the chosen lifts form a topological basis and that no higher Tor contributes when passing to AOmega_S tensor^L A_cris. Since Lemma 2.7 and Construction 3.1 use AOmega_S tensor^L A_cris as an ordinary ring concentrated in degree 0, this gap directly affects the existence of h_crys.
  2. [§2.4, Lemma 2.7 and Remark 2.8] The verification that beta is a PD-thickening is the core local computation, but the proof is a sketch. After 'ker(can) is generated by xi' the conclusion 'x in (phi^{-1}(xi)) subset AOmega_S/(p,xi)' refers to the wrong quotient: the diagram works in AOmega_S/(p,xi^p). The subsequent expression x = phi^{-1}(xi)y + pz' + xi w and the substitution y = p^{(p-1)/p}y' + x_1 treat elements of S and of AOmega_S interchangeably and use phi^{-1}(xi) without specifying how this element is obtained. The 'repeat this procedure' step is not formalized, and convergence or termination of the iteration is not discussed. Remark 2.8 acknowledges that the explicit computation is omitted, but for the construction of h_crys this computation is exactly what must be supplied.
  3. [§3.1, Lemma 3.2] The comparison between h_crys and h_can after base change reduces to a polynomial algebra and then to a perfectoid S. The proof says that by [BdJ11] the two maps agree, but the independence of the choice of the lift Sigma -> tilde{S}/p is not shown. If different lifts give different maps, the resulting h_S is not functorial and the limit in Construction 3.4 would be ill-defined. The proof should spell out why the crystalline-site construction makes h_S independent of auxiliary choices, or should give a canonical construction that does not depend on such choices.
minor comments (5)
  1. [§1.2] There is a typo in the paragraph before Theorem 1: 'our first first main result' should read 'our first main result'.
  2. [§2.4, diagram in Lemma 2.7] The vertical map labelled 'omega |-> omega^p' and the implication alpha(x)=0 implies x^p=0 in LOmega(S/p) need a reference or a one-line justification; both are used without comment.
  3. [§3.2, Construction 3.4] The identification RGamma_Ainf(X) tensor^L A_cris is isomorphic to the homotopy limit of AOmega_R tensor^L A_cris over affine opens Spf R of X_et is asserted without spelling out the required etale descent for the A_inf-cohomology sheaf; a brief justification or reference should be added.
  4. [§4.3, Lemma 4.6] The notation B_cris^+ is used before it is defined in the text; it should be introduced explicitly, or a standard reference for the notation should be given.
  5. [§5.2, Lemma 5.4] The phrase 'the derived quotient Sigma_dR(.)/xi is isomorphic to Sigma_C(.)' would be clearer as a sentence about the termwise derived quotient of the cosimplicial ring Sigma_dR(.), since the individual terms are not all flat over B_dR^+.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction of hcrys is conditional on local PD-thickening lemmas, but those are not the theorems being proved and no fitted parameter or self-citation chain is used.

full rationale

The paper's derivation chain is essentially conditional: Theorem 1 asserts the existence of hcrys, and Theorem 4 is deduced from it. The construction of hcrys in Section 3 reduces, by quasisyntomic descent, to the local statement Lemma 2.7 that for S∈qrsPerfd^proj_{O_C} the map AΩ_S⊗^L A_cris → S/p is a PD-thickening. This lemma is proved using the de Rham comparison (explicitly taken as input, the easier half of [BMS18]), the p-torsion-freeness of S, and the quasi-regularity of S; it does not invoke the crystalline comparison, the B_cris comparison, or the Breuil–Kisin–Fargues structure that the paper later derives. Lemma 2.6 is likewise justified from the freeness of LΩ(S/p) and the discreteness of AΩ on quasi-regular semiperfectoid objects, an external fact from [BMS19], not from the target theorems. There are no fitted parameters and no 'prediction' that is statistically or definitionally forced: hcrys is constructed rather than tuned to match any of the compared cohomologies. The paper's reliance on [BMS19] and [Mor16] is external evidence, not self-citation, since the author is not an author of those works. The abbreviated proofs of Lemmas 2.6 and 2.7, and the omitted explicit computation in Remark 2.8, could be a correctness or rigor concern, but abbreviation is not circularity. The direction of implication is clean: if the local PD-thickening statement holds, the global comparison follows; nothing in the paper assumes the global statement to prove the local one. Therefore no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented physical entities appear. The paper introduces the infinitesimal site Inf(A/B_dR^+) as a formal construction, not an ad hoc entity. All listed axioms are established theorems or explicit inputs from prior literature; the central claim is not assumed.

assumptions (6)
  • domain assumption The de Rham comparison gamma_dR: RΓ_Ainf(X) ⊗^L_{A_inf, θ} O_C → RΓ_dR(X) is taken as an input.
    Section 1.1 states this is the easier part of [BMS18] and is assumed; all later comparisons rely on it.
  • domain assumption Quasiregular semiperfectoid rings form a basis for the quasisyntomic site, and the functor R ↦ AΩ_R ⊗^L A_cris is a sheaf there.
    Used in Lemma 2.4 and Lemma 2.5; the sheaf property is established via [BMS19, Construction 9.5] and is the backbone of the global construction.
  • standard math Derived Nakayama applies to the A_cris to W(k) specialization.
    Corollary 3.7 uses it to lift the isomorphism after base change to k to a quasi-isomorphism over W(k).
  • domain assumption Morrow's Lemma A.4 and Fargues' classification of Breuil-Kisin-Fargues modules.
    Theorem 4.2 and Lemma 4.10 are imported for proving H^i_Ainf(X)[1/p] is finite free and for the reconstruction in Theorem 5.9.
  • domain assumption Berthelot-Ogus comparison for crystalline cohomology over A_cris, Lemma 4.6.
    Attributed to [BMS18]; identifies H^i_crys(X_{O_C}/p/A_cris)[1/p] with H^i_crys(X_k/W(k)) ⊗ B_cris^+.
  • domain assumption Scholze's B_dR Poincaré lemma and filtered pro-etale comparison.
    Used in Lemma 5.13 and Theorem 5.14 for the filtration compatibility of the B_cris isomorphism.

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Pith. "Pith review of The crystalline comparison of Ainf-cohomology: the case of good reduction." pith.science (2026). https://pith.science/paper/IO3NPDJB

@misc{pith2026190806366,
  author       = {Pith},
  title        = {Pith review of: The crystalline comparison of Ainf-cohomology: the case of good reduction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IO3NPDJB}},
  note         = {Machine review of arXiv:1908.06366}
}
read the original abstract

We provide a simple approach for the crystalline comparison of Ainf-cohomology, and reprove the comparison between crystalline and p-adic etale cohomology for formal schemes in the case of good reduction.

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