Galois Cohomology for Lubin-Tate $(\varphi_q,\Gamma_{LT})$-modules over Coefficient rings

Chandrakant Aribam; Neha Kwatra

REVIEW 3 major objections 6 minor 24 references

Galois Cohomology for Lubin-Tate $(\varphi_q,\Gamma_{LT})$-modules over Coefficient rings

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Galois cohomology of local representations is computed by Lubin-Tate Herr complexes, and the equivalence extends to coefficient rings.

desk verdict The coefficient-ring extension of the Lubin-Tate Herr machine is genuinely worth engaging; the False-Tate section is not yet supported, and one lemma in Section 3 is false as stated. read the letter →

arxiv 1908.03941 v4 pith:V4XRWESQ submitted 2019-08-11 math.NT

Chandrakant Aribam , Neha Kwatra This is my paper · ORCID

classification math.NT MSC 11F8011F8511S2511S3114F30

keywords GaloisrepresentationsLocalfieldscohomologyLubin-Tateextensions(phi_qGamma_LT)-modulesCoefficientringsIwasawap-adicformalgroups

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 establishes that Galois cohomology of local Galois representations can be read off from an explicit complex attached to the representation, rather than from the full Galois group. The main theorem states that for a discrete $\pi$-primary representation $V$ of $G_K$, the groups $\mathrm{H}^i(G_K,V)$ are naturally isomorphic to the cohomology of the Lubin-Tate Herr complex $\Phi\Gamma^{\bullet}_{LT}(D_{LT}(V))$ for every $i\ge 0$, and a limit argument extends this to $\mathcal{O}_K$-linear representations. The same construction is adapted to non-abelian False-Tate type extensions, a $\psi_q$ variant is shown to compute Iwasawa cohomology, and the classification of representations by étale $(\varphi_q,\Gamma_{LT})$-modules is extended to coefficient rings. All of this matters because it turns abstract Galois cohomology into a concrete algebraic object that can be manipulated and computed.

What carries the argument

The load-bearing object is the Lubin-Tate Herr complex $\Phi\Gamma^{\bullet}_{LT}(M)$, defined as the total complex of the double complex $\Gamma^{\bullet}_{LT}(\Phi^{\bullet}(M^{\Delta}))$. Here $\Phi^{\bullet}$ is the two-term complex $0\to M \xrightarrow{\varphi_M-\mathrm{id}} M\to 0$, and $\Gamma^{\bullet}_{LT}$ is the Koszul complex whose differentials are $\gamma_i-\mathrm{id}$ for topological generators $\gamma_i$ of $\Gamma^*_{LT}$, the torsion-free quotient of the Lubin-Tate Galois group. The combination converts the semi-linear Frobenius and the action of $\Gamma^*_{LT}$ into the cohomology of the full absolute Galois group $G_K$. The companion machinery is the equivalence $D_{LT}$ (and its coefficient-ring analogue $D_R$) between representations and étale $(\varphi_q,\Gamma_{LT})$-modules, which supplies the module $M$ from a representation $V$. The False-Tate type version replaces $\Gamma^{\bullet}_{LT}$ by a twisted Koszul complex $\Gamma^{\bullet}_{LT,FT}$ whose differentials encode the semidirect product $\Gamma^*_{LT}\rtimes \mathbb{Z}_p$ and the Lubin-Tate character.

What would settle it

Compute $\mathrm{H}^1$ of the twisted complex $\Gamma^{\bullet}_{LT,FT}(A)$ for $A=\mathcal{O}_E/\pi\mathcal{O}_E$ with trivial $\Gamma_{LT,FT}$-action; the theorem requires it to equal $\mathrm{Hom}(\Gamma_{LT,FT}^{\mathrm{ab}},A)$, whose dimension is known from the semidirect product structure. A different answer would refute the unproved step behind Theorem 4.8.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 3.13 and its coefficient-ring version Theorem 8.1: for a discrete $\pi$-primary representation $V$ of $G_K$ there is a natural isomorphism $\mathrm{H}^i(G_K,V)\simeq \mathrm{H}^i(\Phi\Gamma^{\bullet}_{LT}(D_{LT}(V)))$ for $i\ge 0$, where $D_{LT}$ is the functor carrying representations to étale $(\varphi_q,\Gamma_{LT})$-modules and $\Phi\Gamma^{\bullet}_{LT}$ is the Lubin-Tate Herr complex. For $\mathcal{O}_K$-linear representations the same isomorphism follows by passage to inverse limits, and an analogous statement holds for non-abelian False-Tate type extensions. The paper also shows that the $\varphi_q$-complex maps injectively on $\mathrm{H}^0$ into a $\psi_q$-complex and that Iwasawa cohomology is recovered from the $\psi_q$-complex. Over a coefficient ring $R$, the functor $D_R$ is proved to be an equivalence between $R$-linear representations of $G_K$ and étale $(\varphi_q,\Gamma_{LT})$-modules over $O_R=O_E\widehat{\otimes}_{\mathcal{O}_K}R$, and this equivalence carries the cohomological theorems over to $R$.

Load-bearing premise

The argument leans on an unproved assertion in Theorem 4.8 that the twisted non-abelian complex $\Gamma^{\bullet}_{LT,FT}(A)$ computes the group cohomology $\mathrm{H}^i(\Gamma_{LT,FT},A)$ of the semidirect product $\Gamma^*_{LT}\rtimes\mathbb{Z}_p$; the paper says this follows by the same technique as in the abelian case but supplies no proof for that step.

Editorial extensions

If this is right

If the central isomorphisms are correct, Galois cohomology of $\mathcal{O}_K$-linear and $R$-linear representations becomes a finite algebraic computation from $D_{LT}(V)$ or $D_R(V)$, rather than an analysis of the full Galois group.
The vanishing $\mathrm{H}^i(G_K,V)=0$ for $i\ge 3$ becomes a formal consequence of the shape of the Lubin-Tate Herr complex, as the paper notes in Corollary 3.16.
Iwasawa cohomology over the Lubin-Tate tower is governed by the $\psi_q-\mathrm{id}$ complex, and over coefficient rings this yields a dual exponential map $\mathrm{Exp}^*_R$, opening the route to explicit reciprocity maps for $R$-representations.
The equivalence over coefficient rings means that questions about $G_K$-representations over complete local Noetherian rings can be translated into module-theoretic questions about étale $(\varphi_q,\Gamma_{LT})$-modules over $O_R$.
For the non-abelian False-Tate tower, the same cohomological computation works, so the Herr-complex method is not confined to abelian $\mathbb{Z}_p$-extensions.

Reading between the lines

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

If the claims are right, the same double-complex construction should push from Noetherian coefficient rings to $p$-adic Banach or pseudorigid families, since the complex is built from finitely many explicit operators.
The False-Tate case is likely a template for other arithmetically pro-finite extensions whose Galois group is a semidirect product with twisting governed by a Lubin-Tate character; the same twisted Koszul differentials would apply.
Establishing an explicit duality pairing the $\varphi_q$- and $\psi_q$-complexes would make local duality transparent in the Lubin-Tate setting and might yield explicit reciprocity laws for the $R$-valued dual exponential map.

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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

Dear colleague,

Quick take: the paper has two useful halves and one half that is not ready. The genuinely new piece is the coefficient-ring version of Kisin--Ren: Theorems 7.15 and 7.18 extend the étale (phi_q, Gamma_LT)-module equivalence to modules over a complete Noetherian local ring, and Section 8 uses that to carry the Lubin-Tate Herr complex, psi_q-operator, and Iwasawa cohomology to coefficient rings. That part is plausible and, as far as I can tell, not in Dee or in Schneider--Venjakob. The earlier Lubin-Tate Herr complex in Section 3 is a natural extension of Herr, with an understandable overlap with Schneider--Venjakob; the direct-limit formalism is clean and the main comparison Theorem 3.13 has the right shape.

Now the soft spots, in order of severity.

First, Lemma 3.2 is false as written. It says a finite pi-power torsion OK-module is free over OK/pi^n. That is not true; take OK/pi itself. The proof uses flatness of V over OK/pi^n, which is exactly what fails. The quasi-isomorphism may still be provable by a devissage through pi-torsion, but the argument as printed has a real gap. This is fixable, not fatal, but it is in the foundation of Theorem 3.13.

Second, and more serious, is Theorem 4.8 and the whole False-Tate complex. The differentials in Definition 4.6 divide by expressions like (gamma_tilde^a - 1), and on discrete pi-primary modules that operator has nonzero kernel already for trivial modules. If the intended meaning is to invert only on the image, that is not stated, and the degree-one term is A^{d+1}, not the image. Moreover, Gamma_{LT,FT} is a nonabelian semidirect product, so the spectral sequence proof of Proposition 3.8 cannot be copied without additional work. The paper's assertion that this is routine is not credible. This is a load-bearing advertised result, not a minor typo.

Third, Theorem 8.3 relies on an inverse-limit step that is sketched rather than demonstrated. This is less troubling than the False-Tate gap, but the passage from finite-length modules to general R-representations deserves a few lines of proof.

Who is this for? Arithmetic geometers and Iwasawa theorists working with Lubin-Tate extensions and families of Galois representations. They should read the coefficient-ring sections with interest and treat Section 4 with suspicion. The paper deserves a serious referee, but with a request to fix Lemma 3.2, rewrite Section 4 with a well-defined complex and a real proof, and expand the inverse-limit arguments in Section 8. I would not desk-reject it.

Best, [You]

Referee Report

3 major / 6 minor

Summary. The paper develops a Lubin-Tate analogue of Herr's cohomological formalism. It defines a Lubin-Tate Herr complex for (φ_q, Γ_LT)-modules and proves comparisons H^i(G_K,V) ≅ H^i(ΦΓ•_LT(D_LT(V))) for discrete π-primary and O_K-linear representations (Theorems 3.13 and 3.15). It then introduces a False-Tate type extension Γ_LT,FT = Γ*_LT ⋉ Z_p and a corresponding complex ΦΓ•_{LT,FT}, claiming the same comparison (Theorem 4.8). The paper also relates φ_q and ψ_q cohomology (Theorems 5.6, 5.9), computes Iwasawa cohomology in terms of ψ_q (Theorem 6.2), and extends the Kisin--Ren equivalence to coefficient rings R (Theorems 7.15, 7.18), with cohomological consequences over R (Theorems 8.1--8.6). The main tools are categorical dévissage, inverse limits, and spectral sequences.

Significance. If the central theorems hold, the paper gives a systematic way to compute Galois cohomology over Lubin-Tate extensions, including a nonabelian False-Tate analogue and a coefficient-ring version of the Kisin--Ren classification. The paper has several strengths: it builds on clearly stated external results (Kisin--Ren, Dee, Schneider--Venjakob), it constructs explicit complexes (e.g., Examples 3.11, 4.7), and it states parameter-free natural isomorphisms. The coefficient-ring extension via finite-length reduction and inverse limits is a natural and potentially useful strategy. However, the manuscript contains several load-bearing proof gaps that must be repaired before the advertised theorems can be considered established.

major comments (3)

[Lemma 3.2] The proof asserts that a finite abelian π-group V is a free O_K/π^n-module. This is false: for example, V = O_K/π is killed by π^2 but is not free over O_K/π^2. The identity Φ•(D_sep) = Φ•(Ô_Eur/π^n) ⊗_{O_K/π^n} V is valid, but tensoring the quasi-isomorphism (3.1) with V is only exact when V is flat over O_K/π^n. For a general finite π-group one needs to filter V by O_K/π-submodules and use dévissage. Since Lemma 3.2 is used in Proposition 3.4 and hence in Theorem 3.13, this proof gap must be fixed.
[Section 4, Definition 4.6 and paragraph after Example 4.5] The claim that the complex Γ•_{LT,FT}(A) computes H^i(Γ_LT,FT,A) is asserted to follow "using the similar technique as in the proof of Proposition 3.8", but no proof is given. More seriously, the differentials in Definition 4.6 contain expressions such as (γ_j − γ̃^{χ(j)χ(...)} − id)/(γ̃^{χ(...)} − id). On a discrete π-primary module, γ̃ − id need not be invertible; for the trivial module A = Z/p it has a nonzero kernel. Unless the division is reinterpreted as a map defined only on the image of γ̃ − id (which is not stated), the maps are not well-defined endomorphisms of A. Because Γ_LT,FT is nonabelian, the abelian spectral-sequence argument of Proposition 3.8 cannot be copied verbatim. Consequently the well-definedness of ΦΓ•_{LT,FT} and the isomorphism in Theorem 4.8 are not established.
[Theorem 8.3, proof] The proof reduces to the finite-length case and then invokes Corollary 6.3, which applies to V ∈ Rep_{O_K}(G_K), i.e., finite free O_K-modules. For a finite-length R-representation, Lemma 7.9 only provides finite generation over O_K, not freeness; such a representation is generally a finite O_K-torsion module. The correct reference is Theorem 6.2, which treats V ∈ Rep^dis_{O_K-tor}(G_K). The argument is likely repairable, but as written the cited step does not justify the conclusion.

minor comments (6)

[Introduction and Section 3] The assumption that p is odd is introduced only in Section 3; if p = 2 is excluded it should be stated in the introduction, since earlier sections discuss arbitrary p.
[Definition 3.10 and Definition 4.6] The total complex notation Γ•_{LT}(Φ•(M^Δ)) and Γ•_{LT,FT}(Φ•(M)) should explicitly say that the complex Γ• is applied termwise to the two-term complex Φ•. This is standard but should be stated to avoid ambiguity.
[Example 4.5] There are typos in the displayed matrices: an unbalanced parenthesis in the expression for A_2 and a typo "˜γaa2−id" instead of "˜γa1a2−id".
[Section 7.2.1] The notation E is reused for a local field of characteristic p and earlier for the residue field of O_E; this is confusing. Consider using a different letter for the characteristic-p field.
[Theorem 5.6, proof] The assertions H^0(K)=0 and H^0(C)=0 are stated without proof. They require justification for a π-power torsion module with an action of Γ_LT, particularly because the kernel of ψ_M can interact nontrivially with the Γ-action.
[Introduction and references] The introduction attributes the False-Tate extension of Herr's complex to [22] and writes "Floric"; the reference [22] is by Tavares Ribeiro. Please correct the attribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation chain rests on external theorems (Kisin-Ren, Herr, Schneider-Venjakob, Dee) and inverse-limit reductions; the False-Tate gap is a correctness issue, not a circular dependence.

full rationale

The paper does not fit parameters to data, rename a known result, or justify a central premise by a load-bearing self-citation. The main Lubin-Tate Herr complex is defined independently as a double complex built from (phi_q - id) and Koszul-type operators (gamma_i - id), and Theorem 3.13 is proved by comparing two universal delta-functors via spectral sequences and dimension shifting, using the external Kisin-Ren equivalence (Theorem 2.4) and the standard cohomological vanishing of Herr's cyclotomic argument. The coefficient-ring results in Section 7 are proved by reduction to finite-length modules and then by inverse limits, not by assuming the desired equivalence. The proof of Theorem 7.18 explicitly reduces to Theorem 7.15 and external results. The only serious weakness is in Section 4: the claim after Example 4.5 that the Gamma^bullet_{LT,FT}(A) complex computes the nonabelian group cohomology of Gamma_{LT,FT} is asserted without proof, and the differentials in Definition 4.6 formally divide by (gamma-tilde - id), which is not invertible on discrete pi-primary modules. That is a possible mathematical gap or error in a central claimed result, but it is not circularity: the False-Tate complex is not defined in terms of the cohomology it is supposed to compute, and no fitted or self-referential input is being renamed as a prediction. The paper is self-contained against external benchmarks, so the circularity burden is low and the appropriate score is 0.

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

The central results rest on the Kisin-Ren equivalence, the field-of-norms identification, and standard continuous cohomology of Z_p^d. No free parameters or invented entities appear. The main risky input is the unproved False-Tate-type complex assertion.

assumptions (5)

domain assumption Kisin-Ren equivalence between Rep_OK(G_K) and etale (phi_q,Gamma_LT)-modules over O_E (Theorem 2.4 from [11])
Used in Propositions 3.1, 3.4 and throughout; if false, the Herr complex construction has no input.
domain assumption Field-of-norms identification H_K isomorphic to Gal(E^sep/E) (Lemma 2.2 from [11])
Used to identify D_LT(V) invariants and to transfer cohomology triviality from H_K to G_E.
standard math The complex Gamma^bullet_LT(A) computes continuous cohomology of a discrete pi-primary A for Gamma^*_LT isomorphic to Z_p^d (Proposition 3.8)
Proof is sketched via injectives and Hochschild-Serre; the result is standard for Z_p^d acting on discrete p-primary modules.
ad hoc to paper The complex Gamma^bullet_LT,FT(A) computes H^i(Gamma_LT,FT,A) for the semidirect product Gamma_LT,FT
Asserted without proof after Definition 4.6 in Section 4; Theorem 4.8 depends on it.
standard math H^i(H_K,O^hat_E_ur/pi^n)=0 for i at least 1 (Lemma 3.3)
Reduces to additive Galois cohomology of a separably closed field, a standard fact.

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Pith. "Pith review of Galois Cohomology for Lubin-Tate $(\varphi_q,\Gamma_{LT})$-modules over Coefficient rings." pith.science (2026). https://pith.science/paper/V4XRWESQ

@misc{pith2026190803941,
  author       = {Pith},
  title        = {Pith review of: Galois Cohomology for Lubin-Tate $(\varphi_q,\Gamma_LT)$-modules over Coefficient rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V4XRWESQ}},
  note         = {Machine review of arXiv:1908.03941}
}

abstract

The classification of the local Galois representations using $(\varphi,\Gamma)$-modules by Fontaine has been generalized by Kisin and Ren over the Lubin-Tate extensions of local fields using the theory of $(\varphi_q,\Gamma_{LT})$-modules. In this paper, we extend the work of (Fontaine) Herr by introducing a complex which allows us to compute cohomology over the Lubin-Tate extensions and compare it with the Galois cohomology groups. We further extend that complex to include certain non-abelian extensions. We then deduce some relations of this cohomology with those arising from $(\psi_q,\Gamma_{LT})$-modules. We also compute the Iwasawa cohomology over the Lubin-Tate extensions in terms of $\psi_q$-operator acting on the \'{e}tale $(\varphi_q,\Gamma_{LT})$-module attached to the local Galois representation. Moreover, we generalize the notion of $(\varphi_q,\Gamma_{LT})$-modules over the coefficient ring $R$ and show that the equivalence given by Kisin and Ren extends to the Galois representations over $R$. This equivalence allows us to generalize our results to the case of coefficient rings.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Multivariable Lubin-Tate (ϕ, Γ)-modules and ﬁlteredϕ-modules

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Berger, L., Schneider, P., and Xie, B. Rigid Character Groups, Lubin-Tate Theory, and (ϕ,Γ)-Modules. Mem. Amer. Math. Soc. 263, 1275 (2020), v + 79. 2

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