Finite-coefficient Gersten injectivity fails in ramified mixed characteristic

Niels Feld

Finite-coefficient Gersten injectivity fails in ramified mixed characteristic

T0 review · 0 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper constructs a nonzero class in $K_2(A;\mathbb{Z}/3)$ for the complete two-dimensional ramified regular local ring $A=V[[x,y]]/(3+x^2-y^3)$ whose restriction to the fraction field is zero, showing that the termwise…

desk verdict Explicit ramified mixed-characteristic counterexample to finite-coefficient Gersten injectivity; proof is credible but depends on unverified motivic preprints. read the letter →

arxiv 2608.05005 v1 pith:ZSGTEIFX submitted 2026-08-05 math.KT

Niels Feld This is my paper · ORCID

classification math.KT MSC 19E0819E1514C3513H05

keywords GerstenconjecturealgebraicK-theoryfinitecoefficientsmixedcharacteristicramifiedregularlocalringmotivicfiltrationBeilinson-LichtenbaumK_2

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 constructs an explicit counterexample to Gersten’s conjecture with finite coefficients. For $V$ a complete discrete valuation ring of mixed characteristic $(0,3)$ in which $3$ is a uniformizer — concretely $V$ can be $\mathbb{Z}_3$ — the complete ramified regular local domain $A=V[[x,y]]/(3+x^2-y^3)$ carries a nonzero class $a\in K_2(A;\mathbb{Z}/3)$, the mod-3 algebraic $K$-theory group, whose image in $K_2(F;\mathbb{Z}/3)$ for $F=\operatorname{Frac}(A)$ is zero. The class is supported on the divisor $y=0$ and becomes, after adjoining a cube root of unity and inverting $3$, a product of a mod-3 class with a line bundle of order three, detected in weight-two motivic cohomology. The coefficient Bockstein of $a$ is zero, so the failure is not a contradiction to integral Gersten injectivity but rather a genuine $3$-torsion gap in the cokernel of the injective map $K_2(A)\to K_2(F)$. If correct, it rules out a naive term-by-term reduction of Gersten’s conjecture to finite coefficients.

What carries the argument

The mechanism that carries the argument is the multiplicative motivic filtration on algebraic $K$-theory together with the weight-two Beilinson–Lichtenbaum comparison, packaged as Lemmas 2.1 and 2.2. Lemma 2.2 is the central detection statement: on a regular one-dimensional affine scheme $T$ containing a primitive cube root $\zeta$, if $L$ has nontrivial class in $\operatorname{Pic}(T)/3$ and $\beta\in K_2(T;\mathbb{Z}/3)$ has boundary $[\zeta]$, then $\beta([L]-1)\ne 0$. The proof promotes $\beta$ and $[L]-1$ to classes in the first layers of the motivic filtration, multiplies them into the second layer, and reads off the nonzero étale cup product $\zeta\smile\kappa(L)$. The geometric input is the cyclotomic $A_2$ surface $B=O[[x,y]]/(uv-y^3)$, whose divisor class group is cyclic of order three generated by the prime $P=(u,y)$, so $L=P|_T$ has order three and the two conjugate points $P,Q$ make the pullback of $a$ a Bott product (a product of a mod-3 class with a line-bundle class).

What would settle it

One could settle the nonvanishing by computing the étale cup product $\zeta\smile\kappa(L)$ in $H^2_{\mathrm{et}}(T,\mu_3^{\otimes 2})$ for $T=\operatorname{Spec}(B[1/3])$ and $L=P|_T$; since $\kappa(L)$ is the Kummer boundary of a generator of $\operatorname{Pic}(T)\simeq\mathbb{Z}/3$, showing this cup product is zero, or producing an explicit relation killing $\beta|_T([L]-1)$ in $K_2(T;\mathbb{Z}/3)$, would refute Theorem 1.1.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for $V$ as above, the ring $A=V[[x,y]]/(3+x^2-y^3)$ is a complete two-dimensional ramified regular local domain, and there exists $0\ne a\in\ker(K_2(A;\mathbb{Z}/3)\to K_2(F;\mathbb{Z}/3))$ with $\partial_A(a)=0$. The proof isolates $a$ as the pushforward along $\operatorname{Spec}(O)\hookrightarrow\operatorname{Spec}(A)$ of a mod-3 class $\beta$ on $O=V[\zeta]$, where $\zeta$ is a primitive cube root of unity; after the flat base change $A\to B[1/3]$ with $B=O[[x,y]]/(uv-y^3)$, the pullback of $a$ equals $\beta|_T([L]-1)$ on $T=\operatorname{Spec}(B[1/3])$, where $L$ generates $\operatorname{Pic}(T)\simeq\mathbb{Z}/3$. Lemma 2.2 shows that such a product is nonzero because its associated graded class is the motivic cup product $\zeta\smile c_1^{\mathrm{mot}}(L)\bmod 3$, which maps under Beilinson–Lichtenbaum to the nonzero étale class $\zeta\smile\kappa(L)$. Since $a$ is supported on a divisor, its restriction to $F$ is zero; since $A$ contains no nontrivial cube root of unity, its Bockstein vanishes. Thus the termwise finite-coefficient Gersten row, obtained by applying the mod-3 Moore spectrum term by term to the Cousin resolution, is not exact at its first term, while the integral map $K_2(A)\to K_2(F)$ remains injective.

Load-bearing premise

The proof that the constructed class is nonzero depends on a comparison between motivic and étale cohomology, supplied by recent preprints, being valid for the one-dimensional scheme $T=\operatorname{Spec}(B[1/3])$ in this ramified mixed-characteristic setting; if that comparison fails, the counterexample is unsupported.

Editorial extensions

If this is right

For the ring $A$, the augmented termwise finite-coefficient Gersten row for $K$-theory with $\mathbb{Z}/3$-coefficients fails to be exact at its first term.
The integral Gersten conjecture is not contradicted: the map $K_2(A)\to K_2(F)$ is injective, and the obstruction is a nonzero element of the cokernel that is $3$-torsion.
The image of $K_2(A)\to K_2(F)$ is not $3$-saturated: there is an integral class $c$ whose restriction to $F$ is $3d$ for some $d\in K_2(F)$, but no integral class lifts $d$.
The same construction is expected to give, for every odd prime $p$, a nonzero class in $K_{p-1}(A_p;\mathbb{Z}/p)$ vanishing over $\operatorname{Frac}(A_p)$, where $A_p=V[[x,y]]/(x^2-d-y^p)$.
Broad literal formulations of Gersten or Cousin conjectures for arbitrary cohomology theories on all regular schemes are false if they are read as including termwise mod-$p$ algebraic $K$-theory in residue characteristic $p$; the paper notes that the actual theorems in the cited references are unaffected.

Reading between the lines

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

The failure is tied to ramification together with roots of unity: the class is invisible in $A$ until a cube root of unity is adjoined, which suggests testing finite-coefficient Gersten injectivity under hypotheses that exclude nontrivial $p$-th roots of unity in the relevant étale extensions.
The termwise Moore-spectrum construction mixes $K_n(R)/p$ with $K_{n-1}(R)[p]$, so a correct finite-coefficient Gersten statement should probably be formulated as a statement about derived $p$-reduction of one fixed integral Gersten complex rather than about the termwise row.
For $p=3$, one could try to bypass the motivic machinery entirely by writing $\beta$ as an explicit Steinberg symbol in $K_2(O;\mathbb{Z}/3)$ and tracing it through the transfer; such a computation would give an independent check of Theorem 1.1.
If the same mechanism transfers to other cohomology theories with a weight-two motivic/étale comparison, the counterexample may also block literal finite-coefficient Gersten statements for étale or syntomic cohomology in ramified mixed characteristic.

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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

Feld's paper deserves a serious referee. It gives an explicit counterexample to finite-coefficient Gersten injectivity for a two-dimensional ramified regular local ring in mixed characteristic (0,3), which is new and which narrows what the programmatic conjectures in [Kun24, Lüd24] can claim.

The construction is concrete: A = V[[x,y]]/(3+x^2-y^3), and a class a in K2(A;Z/3) that dies in K2(F;Z/3). I checked the local ring arithmetic: A is complete, regular, ramified, special fibre the cusp. The class is pushed forward from the divisor A/yA ≅ V[ζ], then after adding ζ the class becomes a product of a mod-3 class with an order-3 line bundle, detected in weight two of the motivic filtration. Nonvanishing is reduced to Lemma 2.2, and Proposition 4.3 checks the hypotheses carefully, including geometric regularity via Popescu's theorem. The Bockstein vanishing is independent and clean, and the saturation-shadow corollary is a nice payoff. The paper does not overclaim: odd-prime variants are labeled expected work, and Remark 5.4 admits an elementary p=3 route that is not supplied.

The soft spot is the one the paper itself presents: the nonvanishing proof of a depends on recent preprints [Bou24], [Bou25], [BK25], [EM23] for the multiplicative motivic filtration and, in Lemma 2.2, on the Beilinson-Lichtenbaum comparison H^2_mot(T,Z/3(2)) ≅ H^2_et(T,μ_3^⊗2) from [Bou24, Cor. 5.6]. I could not verify those external results, and they are specifically invoked for a ramified, non-smooth one-dimensional T. If any of those preprints fails in that setting, the theorem as proved is unsupported. That is a real risk, but it is external dependence, not an internal gap. The stress-test note about the μ_3→μ_3^⊗2 isomorphism is a non-issue: choosing ζ makes it legitimate. The citations are appropriate; no circularity.

Who should read this: anyone in algebraic K-theory or motivic cohomology interested in the Gersten conjecture. If the motivic inputs survive, it's an important counterexample. If not, it's an unproven claim. Still, the proof structure is good enough that a serious referee should engage.

Referee Report

0 major / 4 minor

Summary. The paper constructs a nonzero class $a \in K_2(A;\mathbf{Z}/3)$ for the complete two-dimensional ramified regular local domain $A = V[[x,y]]/(3+x^2-y^3)$ over a mixed-characteristic $(0,3)$ discrete valuation ring $V$, and shows that $a$ maps to zero in $K_2(F;\mathbf{Z}/3)$, where $F = \mathrm{Frac}(A)$. Thus the termwise finite-coefficient Gersten row is not exact at its first term for this ring. The proof proceeds by pushing forward a class from the divisor $y=0$, then detecting its nonvanishing after a flat base change to a cyclotomic cover, using the multiplicative motivic filtration on $K$-theory and a Beilinson-Lichtenbaum comparison to étale cohomology. The coefficient Bockstein of $a$ is shown to vanish, and the paper also sketches an expected analogue for every odd prime, explicitly labeled as not fully proven.

Significance. If correct, Theorem 1.1 is a significant counterexample: it refutes the literal statement of finite-coefficient Gersten injectivity for ramified regular local rings in mixed characteristic, while leaving the integral Gersten conjecture and the smooth mixed-characteristic cases untouched. The example is explicit and the nonvanishing argument is structurally detailed. The paper also makes a useful conceptual point: the termwise finite-coefficient Gersten row mixes a quotient of the integral row with torsion from one degree lower, so it is not the derived mod-$p$ reduction of a single integral row. The proof leans on very recent motivic-filtration results, which is both a strength (state-of-the-art technique) and a verification risk, but the internal logic is coherent and the cited results are publicly available.

minor comments (4)

[Proposition 4.3] The sentence 'Both identifications $B/P\cong O\cong B/Q$ restrict to the identity on $O$, so $i_{P*}(\beta|_T)=i_{Q*}(\beta|_T)=\beta|_{K_O}$' is a typo: the left-hand side is a class in $K_2(R;\mathbf{Z}/3)$ while the right-hand side is a class in $K_2(K_O;\mathbf{Z}/3)$. The intended statement is the projection formula $i_{P*}(\beta|_{K_O})=\beta|_T\cdot [R/PR]$ and similarly for $Q$.
[Sections 1 and 4] The nonvanishing proof of $a$ depends on the Beilinson-Lichtenbaum comparison $H^2_{\mathrm{mot}}(T,\mathbf{Z}/3(2))\cong H^2_{\mathrm{et}}(T,\mu_3^{\otimes 2})$ from [Bou24, Cor. 5.6] and on the multiplicative motivic filtration from [EM23, Bou24, Bou25, BK25]. I recommend that the authors add an explicit sentence in the introduction or in Section 4 stating that Theorem 1.1 is conditional on these cited preprints, or better, include the elementary symbol computation for $p=3$ mentioned in Remark 5.4 to make the paper self-contained.
[Lemma 2.2] The notation $\beta([L]-1)$ for the product of $\beta\in K_2(T;\mathbf{Z}/3)$ with $[L]-1\in K_0(T)$ is slightly misleading; consider writing $\beta\cdot([L]-1)$ or $\beta\cdot\lambda$ to avoid any possible confusion with evaluation.
[Abstract and Section 1] The phrase 'expected analogous construction for every odd prime' is accurate, but the abstract could more precisely say 'we sketch an expected analogue' so that the speculative status is immediately clear to the reader.

Circularity Check

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No circularity: the counterexample is built from an elementary divisor pushforward, with nonvanishing supplied by independent motivic-comparison results.

full rationale

No circular step is present. The class a is constructed explicitly as the pushforward of a mod-3 class β from the divisor A/yA ≅ O, and its generic restriction vanishes by Quillen's localization sequence. Nonvanishing is obtained by flat base change to R = B[1/3], where the g∗(a) is rewritten as β|T([L]−1), and Lemma 2.2 is then applied to T after checking its hypotheses in Proposition 4.3: T is integral, regular, one-dimensional, invertible over Z/3, contains ζ, has Pic(T) ≅ Z/3 generated by L, and satisfies ∂T(β|T) = [ζ]. Lemma 2.2 itself proves β([L]−1) ≠ 0 from the weight-two motivic cup product ζ⌣(c1^mot(L) mod 3), identified with the étale cup product ζ⌣κ(L) via the Beilinson-Lichtenbaum isomorphism cited from [Bou24, Cor. 5.6]. That cited comparison is an external result by different authors, not by Niels Feld, and it does not assume the Gersten-injectivity failure being established. The other external inputs, the multiplicative motivic filtration from [EM23, Bou24, Bou25, BK25], are likewise independent of the target statement. The Bockstein vanishing is proven separately from K1(A)[3] = 0, which follows from the degree-two field extension F(ζ)/F and does not rely on the motivic machinery. There are no fitted parameters, no prediction read back from data, and no load-bearing self-citation. The paper's dependence on recent preprints is reliance on the literature, not circularity; if those preprints are wrong, the proof would be unsupported, but that is a correctness risk, not a circularity defect.

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

The central claim rests on the motivic filtration machinery from several recent preprints, so the ledger is dominated by domain assumptions from that work. There are no fitted free parameters, and the only new mathematical object, the class a, is explicitly constructed rather than postulated as an explanatory entity.

assumptions (5)

domain assumption Multiplicative motivic filtration on K-theory for one-dimensional regular affine schemes ind-smooth over characteristic zero, with the properties stated in Lemma 2.1.
Invoked at the start of Lemma 2.1 via [Bou24, Defs. 3.18-3.20, Prop. 4.46], [Bou25, Constr. 4.3], [BK25, Thm. 6.1]. These are recent preprints, not proven in this paper.
domain assumption Beilinson-Lichtenbaum isomorphism H^2_mot(T,Z/3(2)) to H^2_et(T,mu_3^{x2}) for T with 3 invertible.
Used in Lemma 2.2 to detect nonzero cup products; cited from [Bou24, Cor. 5.6].
standard math Popescu's theorem: a geometrically regular algebra over a perfect field is a filtered colimit of smooth algebras.
Used in Proposition 4.3 to justify the ind-smooth hypothesis for T; cited from Stacks Project Tag 07GC.
standard math Van der Kallen's theorem on injectivity of K_2(A) -> K_2(Frac(A)) for two-dimensional regular local rings.
Used in Corollary 5.1 to interpret the counterexample via non-saturation; cited [vdK76].
standard math Quillen localization sequence and resolution theorem for K-theory with finite coefficients.
Used in Proposition 4.3 to identify G-groups with K-groups and to kill a on A[1/y]; cited [Qui73, Wei13].

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Cite this review

Pith. "Pith review of Finite-coefficient Gersten injectivity fails in ramified mixed characteristic." pith.science (2026). https://pith.science/paper/ZSGTEIFX

@misc{pith2026260805005,
  author       = {Pith},
  title        = {Pith review of: Finite-coefficient Gersten injectivity fails in ramified mixed characteristic},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSGTEIFX}},
  note         = {Machine review of arXiv:2608.05005}
}

abstract

Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,3)$ in which $3$ is a uniformizer, and put $A=V[[x,y]]/(3+x^2-y^3)$. We construct a nonzero class $a\in K_2(A;\mathbf Z/3)$ whose restriction to the fraction field of $A$ is zero. Thus Gersten injectivity for algebraic $K$-theory with $\mathbf Z/3$-coefficients fails for a two-dimensional ramified regular local ring. The coefficient Bockstein of $a$ is zero, while the map $K_2(A)\to K_2(F)$ is injective. We also indicate the expected analogous construction for every odd prime. This counterexample does not contradict the integral Gersten conjecture but it rules out a naive reduction to finite coefficients.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

2025 , eprint =

Bachmann, Tom and Elmanto, Elden and Morrow, Matthew , title =. 2025 , eprint =. doi:10.48550/arXiv.2508.09915 , url =

arXiv doi:10.48550/arxiv.2508.09915 2025

[2] [2]

2025 , eprint =

Bouis, Tess and Kundu, Arnab , title =. 2025 , eprint =. doi:10.48550/arXiv.2506.09910 , url =

arXiv doi:10.48550/arxiv.2506.09910 2025

[3] [3]

2024 , eprint =

Bouis, Tess , title =. 2024 , eprint =. doi:10.48550/arXiv.2412.06635 , url =

arXiv doi:10.48550/arxiv.2412.06635 2024

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Bouis, Tess , title =. 2025 , eprint =. doi:10.48550/arXiv.2507.16501 , url =

work page Pith review arXiv doi:10.48550/arxiv.2507.16501 2025

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2023 , eprint =

Elmanto, Elden and Morrow, Matthew , title =. 2023 , eprint =. doi:10.48550/arXiv.2309.08463 , url =

work page doi:10.48550/arxiv.2309.08463 2023

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Journal of Algebra , volume =

Gillet, Henri , title =. Journal of Algebra , volume =. 1986 , pages =

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Gillet, Henri and Levine, Marc , title =. Journal of Pure and Applied Algebra , volume =. 1987 , doi =

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Kundu, Arnab , title =. 2024 , eprint =. doi:10.48550/arXiv.2404.06655 , url =

arXiv doi:10.48550/arxiv.2404.06655 2024