Une conjecture $C_{\rm st}$ pour la cohomologie \`a support compact

Pierre Colmez; Sally Gilles; Wies{\l}awa Nizio{\l}

REVIEW 4 major objections 4 minor 6 references

Une conjecture $C_{\rm st}$ pour la cohomologie \`a support compact

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Adjoining p-adic logarithms of p and 2πi to the ring B on the Fargues-Fontaine curve kills its Galois cohomology in positive degrees, making a compact-support C_dR/C_st conjecture possible.

desk verdict A short, inventive note that kills Galois cohomology with two logs and sets up a compact-support C_dR/C_st conjecture; the main theorem is believable but rests on an incompletely documented almost étale descent. read the letter →

arxiv 2511.14675 v3 pith:S6P4FZ6Y submitted 2025-11-18 math.AG math.NT

Pierre Colmez , Sally Gilles , Wies{l}awa Nizio{l} This is my paper · ORCID

classification math.AGmath.NT MSC 14F3014G2211S25

keywords p-adicHodgetheoryFargues-FontainecurveGaloiscohomologyperiodringsC_dRconjectureC_stcompactsupportlogarithms

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

This paper proves a vanishing theorem that removes the main obstruction to formulating compact-support versions of the p-adic comparison conjectures. Let B be the ring of analytic functions on the Fargues-Fontaine curve; the paper shows that if one adjoins a p-adic Kummer logarithm of p (log p̃) and a formal logarithm of the p-adic 2πi (log t), the Galois cohomology of B[log p̃, log t] vanishes in degrees ≥1, with H^0 equal to the base field F. The same holds for B[1/t], and for B_dR adjoining log t alone is enough. The reason this matters is that the original C_dR and C_st conjectures work for proper varieties but fail for compact support cohomology because the period rings have nonzero Galois cohomology in degree 1, producing parasitic contributions. Using the new vanishing results, the paper states Conjecture 3.1, a compact-support analogue of C_dR and C_st based on a derived Hom and on the enlarged period ring B[log p̃, log t, 1/t].

What carries the argument

The mechanism is the adjunction of two logarithms. log t is a formal transcendental element with Galois action σ(log t) = log t + log χ_cycl(σ), so it turns the classes K·log χ_cycl into coboundaries. log p̃ is a transcendental element over B satisfying φ(log p̃) = p log p̃ and σ(log p̃) = log p̃ + Kum_p(σ)t, where Kum_p is the Kummer cocycle of p; it kills the F·Kum_p class inside H^1(G_K,B). The proof then uses the filtration by total polynomial degree in log p̃ and log t and the quotient descriptions of B/tB to show by induction that the higher cohomology vanishes and the first-cohomology transition maps are zero. The special quotient structure B/tB ≅ ∏ C is what makes B work, and the pap

What would settle it

Compute H^2(G_K,t^jB) for a concrete case such as K=Q_p and j=0; a single nonzero class would contradict Lemma 2.8 and therefore Theorem 2.4. More broadly, constructing a nonzero 2-cocycle on G_K with values in B whose class survives after adjoining log p̃ and log t would falsify the paper's central claim.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.4: for Λ = B or B[1/t], one has H^i(G_K, Λ[log p̃, log t]) = F for i=0 and 0 for all i≥1. The proof reduces the cohomology of B to the classical cohomology of the completed algebraic closure C via the exact quotient description B/tB ≅ ∏_{n∈Z} C, identifies H^1(G_K,B) as generated by the Kummer class F·Kum_p together with product-of-K classes times log χ_cycl, and then shows by induction on polynomial degree in the two logarithms that all positive-degree classes are killed. The analogous result for B_dR (Theorem 1.4) requires only log t. These vanishings are exactly what is needed for Conjecture 3.1, which replaces the ordinary Hom in the C_dR/C_st isomorphisms

Load-bearing premise

The main vanishing theorem depends on Lemma 2.8, which asserts—with no proof or reference, via a one-sentence appeal to 'almost étale descent'—that H^i(G_K,t^jB)=0 for all i≥2; if that assertion fails, Theorem 2.4 and Conjecture 3.1 collapse.

Editorial extensions

If this is right

The compact-support C_dR conjecture can now be stated with B_dR[log t]; the adjoined log t trivializes the Ext^1 class that would otherwise double-count de Rham cohomology.
The compact-support C_st conjecture must use the Fargues-Fontaine ring B[log p̃, log t, 1/t] rather than B_st[log t], because H^1(G_K,B_st[log t]) does not vanish; with B[log p̃] the vanishing holds.
When the variety is proper, the new formulation recovers the original Hom-based conjectures, so the traditional C_dR/C_st statements are a special case.
Because H^i(G_K,B) → H^i(G_K,B[1/t]) is an isomorphism (Lemma 2.6), inverting t causes no loss of cohomological information, so the vanishing is stable under localization.
The same logarithmic construction does not work for the more classical rings B^+_cris or B^+_rig; the special quotient structure of B/tB is essential.
A single nonzero class in H^2(G_K,t^jB) would contradict Lemma 2.8 and therefore Theorem 2.4; computing this group for a concrete case such as K=Q_p and j=0 would settle the paper's central claim.

Reading between the lines

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

A natural test of Conjecture 3.1 would be to compute both sides for simple non-proper examples, such as the affine line or a punctured curve, where compact-support cohomology and the period-ring Ext groups are explicitly computable; the affine-curve calculation mentioned in the paper is exactly the sort of check that would support or force a modification of the conjecture.
If the unproved almost étale descent assertion behind Lemma 2.8 fails, the compact-support conjecture as stated would need to be reformulated; supplying a proof or a counterexample for that single vanishing statement is the fastest way to test the paper's main conclusion.
The observation that B^+_cris and B^+_rig cannot be fixed by finitely many adjunctions suggests that the compact-support comparison is genuinely tied to the Fargues-Fontaine curve, possibly pointing to a family of period rings parameterized by intervals of the curve for which analogous vanishing and comparison statements could hold.

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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

Colleague,

The thing to know: this is a real result. Theorem 2.4, showing H^i(G_K, B[log p-tilde, log t]) = 0 for i≥1 and F in degree 0, is genuinely new, and the compact-support C_dR/C_st formulation in Conjecture 3.1 is a natural but nontrivial extension. The reader's conditional verdict is about right.

What's new is the vanishing for B itself. The B_dR[log t] analogue is folklore, but B is the Fargues–Fontaine analytic ring, and adjoining both p-adic logs really does kill the positive-degree cohomology. The strategy is coherent: Tate's description of H^1(G_K,C), the Kummer cocycle, and the two log classes kill the two surviving classes. Lemma 2.6 reducing B[1/t] to B is fine, and Lemma 2.7's computation of the image of H^1(G_K,tB) is plausible, though it leans on a cited proposition.

The soft spot is Lemma 2.8. Its proof is one sentence: 'par descente presque étale, on prouve que H^i(H_K,t^jB)=0 pour tout i≥1 et tout j', followed by cohomological dimension of Γ_K. No statement of the descent theorem, no reference, no verification of the hypotheses. This is not cosmetic. Lemma 2.8 provides the H^2 vanishing needed in Proposition 2.9 and the i≥2 vanishing used in Lemma 2.11's induction. Without it, Theorem 2.4 has a hole. I don't think the claim is false—it smells like a standard almost étale descent result in this setting—but a referee cannot verify it from the text. The passage from the Banach rings B_{[r,s]} to the inverse limit B is also undocumented here.

A smaller soft spot: Remark 3.2(iv) asserts that replacing B_st^+ by B[log p-tilde] leaves Conjecture 0.1 unchanged, but no proof is given. The conjecture itself is motivated well, and the explanation of why a naive RHom fails is sensible.

No circularity, no fitted equations. The paper is honest about what is folklore and flags its own limits—for instance, it explains why the same trick does not work for B_cris^+. The value is real: a clean way to kill the Ext^1_G_K terms for de Rham representations, and a concrete conjecture for compact support cohomology.

This deserves a serious referee. The gap in Lemma 2.8 is fixable, but it needs to be closed before the conjecture is fully supported. I'd cite it once I'm confident that descent step holds up.

Recommendation: send to peer review. The main theorem is significant and the proof strategy is sound; the missing details are a repair, not a wreck.

Referee Report

4 major / 4 minor

Summary. The paper studies the Galois cohomology of the period ring B of analytic functions on the Fargues–Fontaine curve (and its localization B[1/t]) and of the de Rham period rings B_dR^+, B_dR. Its main theorem, Théorème 2.4, states that after adjoining formal generators log \tilde p and log t, the positive-degree continuous Galois cohomology vanishes: H^i(G_K, Λ[log \tilde p, log t]) = 0 for i ≥ 1, where Λ = B or B[1/t]. The proof proceeds by computing H^1(G_K, B) via a Kummer class and a cyclotomic log class (Proposition 2.9) and then using a polynomial-degree induction (Lemma 2.11) to show that both classes are killed by the adjoined logarithms. The analogous statement for B_dR[log t] is proved in Théorème 1.4. On this basis, the paper formulates Conjecture 3.1, a compact-support analogue of the C_dR and C_st conjectures, using a derived Hom and the rings B_dR[log t] and B[log \tilde p, log t, 1/t].

Significance. If the main theorem is correct, it removes a well-known obstruction to formulating compact-support p-adic comparison conjectures: the parasitic positive-degree Galois cohomology of period rings. The proposed C_dR/C_st formulation for compact support is natural and is shown to reduce to known statements in the proper case. The paper is concise and the central strategy is transparent: Tate's theorem controls the cohomology of C, and the two logarithms are chosen precisely to eliminate the two H^1 classes. A notable strength is that the vanishing is not assumed; it is derived from Tate's theorem and previously established period-ring results. However, the proof relies on at least one substantial external input—Lemma 2.8's almost étale descent vanishing—that is asserted without proof or reference, and several other cited results are used without verifying their hypotheses. These gaps are load-bearing for the main theorem and for the conjectures that depend on it.

major comments (4)

[§2.2, Lemma 2.8] This lemma asserts H^i(G_K, t^j B)=0 for all j and all i≥2. The proof is one sentence: by 'descente presque étale' one proves H^i(H_K, t^jB)=0 for all i≥1, and then cd(Γ_K)≤1. No statement of the descent theorem, no verification of its hypotheses (e.g., the relevant Banach/perfectoid properties of the H_K-action on B, or the limit from B_{[r,s]} to B) and no reference are supplied. This is not cosmetic: Proposition 2.9 needs H^2(G_K,tB)=0 to obtain the surjection onto H^1(G_K,∏ C), and Lemma 2.11 uses H^i(G_K,Λ)=0 for i≥2 to run the induction that yields the main vanishing. Without Lemma 2.8, Theorem 2.4 and hence the justification of Conjecture 3.1 collapse. A complete proof or a precise reference with all hypotheses checked must be supplied.
[§2.2, Lemma 2.7] The proof of Lemma 2.7—that the image of H^1(G_K,tB) in H^1(G_K,B) is one-dimensional generated by the Kummer class of log \tilde p—relies on [2, prop.10.7], applied to φ^n(c_σ). It is not explained why φ^n(c_σ) satisfies the hypotheses of that proposition, in particular condition (ii) concerning values in B_dR^+. Since Lemma 2.7 is used to describe H^1(G_K,B) in Proposition 2.9 and hence the logarithms that must be adjoined, the missing verification is load-bearing. The authors should either spell out the application of [2, prop.10.7] or give a direct proof.
[§2.1, Proposition 2.2] The injectivity of the GK-equivariant map K⊗_F B[log \tilde p] → B_dR^+ is cited to [4, prop.10.3.15] without checking that the hypotheses of that proposition are satisfied in the present setting. This injectivity is used in Corollary 2.3(ii) to identify H^0(G_K, B[1/t, logt, log\tilde p]) = F, which is part of the i=0 case of Theorem 2.4. The authors should indicate how [4, prop.10.3.15] applies, or provide a self-contained argument.
[§1.3, Théorème 1.4] The proof of Théorème 1.4 is only the sentence 'La preuve de la prop.1.2 peut alors s'adapter'. Since this theorem is also needed for the C_dR part of Conjecture 3.1, and the filtration by t^j B_dR^+/t^{j+1}B_dR^+ ≅ C(j) does not immediately behave like the polynomial filtration in Proposition 1.2 (one must handle the inverse/direct limit and the fact that B_dR is not a direct limit of these graded pieces in an obvious way), the reader deserves at least an outline of the adaptation. This is a minor issue if the folklore statement is indeed standard, but it should be made precise.

minor comments (4)

[§1.2, proof of Prop. 1.2] In the proof of the H^0 statement, the argument 'x:= exp((p−1)p^N a_k)' requires that a_k be sufficiently integral for exp to converge in C; this should be stated explicitly. Also, the notation H0(G_K, C(−j)) on line 3 of p. 3 should be H^0.
[§2.1, notation] The map p^♭ and the element \tilde p = [p^♭] are used without recalling that ♭ is the tilt of C with respect to p; readers unfamiliar with the Fargues–Fontaine formalism may need a reference or one line of explanation.
[§2.1, Lemma 2.6] The equality H^i(G_K,B[1/t]) = lim_{\to j} H^i(G_K,t^{-j}B) is stated as immediate. Since these are continuous cohomology groups of a compact group, this is a standard commutation with direct limits, but a brief justification (e.g., via the bar resolution or the fact that G_K is compact) would improve readability.
[General] The reference list, while appropriate, does not include a source for the 'folklore' B_dR^+ statement discussed in the Introduction. If the authors do not wish to prove it in detail, they should at least cite a location where it appears.

Circularity Check

1 steps flagged · score 2.0 of 10

No circular reduction in the main derivation; the only flagged issue is an unproved almost-étale descent input in Lemma 2.8, a rigor gap rather than a tautology.

other [Lemme 2.8, §2.2 (used in Prop. 2.9 and Lemma 2.11 for Thm. 2.4)]
"Par descente presque étale, on prouve que H^i(H_K, t^j B)=0 pour tout i≥1 et tout j. Donc l'inflation H^i(Γ_K, H^0(H_K, t^jB))→H^i(G_K, t^jB) est un isomorphisme pour tout i≥1 et tout j. Le résultat est donc une conséquence de ce que Γ_K est de dimension cohomologique 1."

This is not a circular reduction: the lemma does not assume Theorem 2.4, and no equation here is equivalent by construction to the vanishing being proved. It is flagged under the reviewing rule because it is an omitted, load-bearing verification: the 'descente presque étale' step is stated without proof or reference, and it supplies the vanishing H^i(G_K,t^jB)=0 (i≥2) needed for the surjectivity in Prop. 2.9 and the induction in Lemma 2.11. An unproved external input is a rigor gap, not a circularity; hence the score remains low.

full rationale

Theorem 2.4 is proved by an explicit induction (Lemma 2.11) starting from Tate's theorem (Prop. 1.1) and period-ring facts. The formal variables log t and log ptilde are introduced with explicit Galois actions (σ(log t)=log t+log χ_cycl, σ(log ptilde)=log ptilde+Kum_p(σ)t), so the Tate and Kummer classes become coboundaries by construction; the proof then verifies that no higher-degree classes survive. This is a standard mathematical construction, not a fitted input or a renaming. The self-citations [2] and [3] are used as established theorems (e.g. [2, prop.10.7] in Lemma 2.7) and do not by themselves force the conclusion; the central induction is carried out in this paper. The only manuscript passage that the reviewing rule requires flagging is Lemma 2.8's one-sentence appeal to 'descente presque étale' with no reference; this is an omitted proof/correctness risk, but it is not a circular step, so the circularity burden is low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The paper's central claim rests on several heavy external theorems (Tate, Fargues-Fontaine, Colmez 2008) and one asserted almost etale descent step. No numbers are fitted to data; the formal logarithms are the only new algebraic objects, and they lack independent falsifiable predictions.

assumptions (5)

standard math Tate's computation of H^i(G_K,C(j))
Unproved background theorem from Tate [5], used throughout Sections 1 and 2 as the base case for all cohomology computations.
domain assumption B/tB is isomorphic to the product over n in Z of C
Lemma 2.1, cited from Bosco [1] and Fargues-Fontaine [4]; foundational for controlling H^1(B) in Proposition 2.9.
domain assumption Injectivity of K tensor_F B[log p-tilde] into B_dR^+
Proposition 2.2, cited from [4, prop.10.3.15]; needed to identify log p-tilde with log(p-tilde/p) in B_dR^+ and to compute H^0.
domain assumption Proposition 10.7 of [2] used in Lemma 2.7
External result from Colmez 2008 that solves cocycles in B up to F[log p-tilde]; load-bearing for the description of the image of H^1(tB) in H^1(B).
domain assumption Almost etale descent: H^i(H_K,t^j B)=0 for i>=1 and all j
Asserted in Lemma 2.8 with no proof or reference; supports Proposition 2.9 and therefore Theorem 2.4.

invented entities (2)

log t
purpose: Formal p-adic logarithm with sigma(log t)=log t+log chi_cycl(sigma); kills the H^1 class of log chi_cycl in B_dR and B.
Introduced as a formal adjunct; no falsifiable handle outside the formalism. The paper notes it is only used to kill parasitic classes and does not appear in surviving cohomology.
log p-tilde
purpose: Formal p-adic analogue of log p with phi(log p-tilde)=p log p-tilde and Galois action via the Kummer cocycle; kills the F.Kum_p class in H^1(B).
Defined as transcendental over B, with an injective image in B_dR^+ by Proposition 2.2; no independent evidence outside the period-ring formalism.

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Cite this review

Pith. "Pith review of Une conjecture $C_{\rm st}$ pour la cohomologie \`a support compact." pith.science (2026). https://pith.science/paper/S6P4FZ6Y

@misc{pith2026251114675,
  author       = {Pith},
  title        = {Pith review of: Une conjecture $C_\rm st$ pour la cohomologie \`a support compact},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S6P4FZ6Y}},
  note         = {Machine review of arXiv:2511.14675}
}

abstract

Let $\mathbf{B}$ be the ring of analytic functions on the Fargues-Fontaine curve $Y_{\rm FF}$. We show that adding $p$-adic analogs of $\log p$ and $\log 2\pi i$ kills its Galois cohomology in degrees~$\geq 1$. The analogous result for $\mathbf{B}^+_{\rm dR}$ is folklore. This makes it possible to formulate $C_{\rm dR}$ and $C_{\rm st}$-type conjectures for compact support cohomology of $p$-adic analytic varieties.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 1 linked inside Pith

[1]

G.\,Bosco , Rational p -adic Hodge theory for rigid-analytic varieties, arXiv:2306.06100v1 [math.AG], preprint 2023

arXiv 2023
[2]

P.\,Colmez , Espaces Vectoriels de dimension finie et repr\'esentations de de Rham, Ast\'erisque 319 (2008), 117--186

2008
[3]

P.\,Colmez, W.\,Nizio , On the cohomology of p -adic analytic spaces, II: The C_ st -conjecture , Duke Math. J. 174 (2025), 2203--2301

2025
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Fargues et J.-M

L. Fargues et J.-M. Fontaine , Courbes et fibr\'es vectoriels en th\'eorie de Hodge p -adique, Ast\'erisque 406 (2018), 51--382

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\@lbibitem[#1]#2 [\@biblabel #1 ] @tempwidthb \@biblabel #1 @tempwidthb> @tempwidtha @tempwidtha= @tempwidthb @filesw \@auxout #2 #1 \@bibitem#1 @filesw \@auxout #1 \@listctr @tempwidthb \@biblabel @tempwidthb> @tempwidtha @tempwidtha= @tempwidthb @multicol multicol.sty \@classname Loading multicol.sty multicol @multicoltrue \@classname multicol.sty is un...

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

G.\,Bosco , Rational p -adic Hodge theory for rigid-analytic varieties, arXiv:2306.06100v1 [math.AG], preprint 2023

arXiv 2023

[2] [2]

P.\,Colmez , Espaces Vectoriels de dimension finie et repr\'esentations de de Rham, Ast\'erisque 319 (2008), 117--186

2008

[3] [3]

P.\,Colmez, W.\,Nizio , On the cohomology of p -adic analytic spaces, II: The C_ st -conjecture , Duke Math. J. 174 (2025), 2203--2301

2025

[4] [4]

Fargues et J.-M

L. Fargues et J.-M. Fontaine , Courbes et fibr\'es vectoriels en th\'eorie de Hodge p -adique, Ast\'erisque 406 (2018), 51--382

2018

[5] [5]

of a conference on local fields , Nuffic Summer School at Driebergen, 158--183, Springer 1967

J.\,Tate , p -divisible groups, Proc. of a conference on local fields , Nuffic Summer School at Driebergen, 158--183, Springer 1967

1967

[6] [6]

\@lbibitem[#1]#2 [\@biblabel #1 ] @tempwidthb \@biblabel #1 @tempwidthb> @tempwidtha @tempwidtha= @tempwidthb @filesw \@auxout #2 #1 \@bibitem#1 @filesw \@auxout #1 \@listctr @tempwidthb \@biblabel @tempwidthb> @tempwidtha @tempwidtha= @tempwidthb @multicol multicol.sty \@classname Loading multicol.sty multicol @multicoltrue \@classname multicol.sty is un...

2000