Torsion in Tate-twisted étale cohomology over $\bar{\mathbb{F}}_p$: is $H^n(X,\mu_{\ell^r}^{\otimes n})$ equal to $N^1H^n$? (Bloch–Esnault's question)

For a smooth projective variety $X$ over the algebraic closure $k=\bar{\mathbb{F}}_p$ of a finite field and a prime $\ell\neq p$, a question attributed to Bloch–Esnault (Math. Ann. 304 (1996), p. 305) and recently highlighted by Schreieder (arXiv:2406.18706, Remark 2.4) asks whether, for all $n,r\geq1$, the Tate-twisted torsion group $H^n(X_{\mathrm{\acute{e}t}},\mu_{\ell^r}^{\otimes n})$ coincides with its coniveau-one part $N^1H^n$, i.e. whether every such torsion class is supported on a divisor. The cases $n=1,2$ are well understood and the equality already fails there in general; for $n\geq3$ the question is open, with no weight-theoretic obstruction available over $\bar{\mathbb{F}}_p$. Schreieder relates the question to Milnor K-theory: his Remark 2.4 records that, for a given variety $X$, the equality for all $r\geq1$ implies that the natural map $H^n(X,\mathbb{Z}_\ell(n))\to K^M_n(k(X))^{\wedge}\otimes\mathbb{Z}_\ell$ is zero (the source states only this one-way implication), and he expects the equality is likely not true in general. Schreieder states that the question is open to the best of his knowledge (Remark 2.4).

Background

Let $X$ be a smooth projective variety over a field $k$ and let $\ell$ be a prime with $\ell\neq\mathrm{char}(k)$. Étale cohomology $H^i(X_{\mathrm{\acute{e}t}},-)$ is the algebro-geometric analogue of singular cohomology; its coefficient systems include the finite groups $\mu_{\ell^r}^{\otimes n}$, the $n$-fold tensor powers of the group of $\ell^r$-th roots of unity (a Tate twist), whose cohomology groups are finite and carry an action of the Galois group of $k$. When $k=\bar{\mathbb{F}}_p$, the algebraic closure of a finite field, there is no Galois action to exploit, but there is a Frobenius (weight) structure on the cohomology of models over the underlying finite field.

The coniveau filtration $N^cH^n(X_{\mathrm{\acute{e}t}},-)$ measures the codimension of support of a cohomology class: $N^cH^n$ consists of the classes that vanish after restriction to the complement of some closed subset of codimension at least $c$, or equivalently of sums of Gysin pushforwards (refined cycle-class maps) from cohomology of codimension-$c$ subvarieties. The quotient $H^n/N^1H^n$ is the unramified cohomology $H^n_{\mathrm{nr}}(k(X)/k,-)$, an invariant of the function field $k(X)$ introduced in this context by Bloch–Ogus and heavily used by Colliot-Thélène–Ojanguren and Bloch–Esnault to detect codimension-2 cycles not algebraically equivalent to zero. Milnor K-theory $K^M_n(F)$ of a field $F$ is the graded ring generated by symbols subject to bilinearity and the Steinberg relation; its completion $K^M_n(k(X))^{\wedge}$ at $\ell$ receives cycle classes and controls the Gersten-type complex relating cycle theory to étale cohomology.

The question studied here sits at the border between the coniveau filtration and the Tate conjecture. The Tate conjecture for divisors over a finite field $\mathbb{F}_q$ predicts that cycle classes of codimension-1 cycles span the Galois-invariant part of $H^2(X_{\bar{\mathbb{F}}_q},\mathbb{Q}_\ell(1))$; it is open in general. Schreieder's paper (arXiv:2406.18706) re-examines the approach to this conjecture through Milnor K-theory and the Bloch–Ogus/Gersten complex, proving (Theorem 1.1) that the Tate conjecture for divisors over $\mathbb{F}_q$ is equivalent to $q$ not occurring as an eigenvalue of the arithmetic Frobenius acting on the completed third Milnor K-group $K^M_3(\bar{\mathbb{F}}(\mathbb{P}^3))^{\wedge}\otimes\mathbb{Q}_\ell$ of the function field of projective three-space. In that framework, Remark 2.4 of the paper records — as a question attributed to Bloch–Esnault's 1996 paper on the coniveau filtration (p. 305) — whether over $k=\bar{\mathbb{F}}$ the groups $H^n(X,\mu_{\ell^r}^{\otimes n})$ have coniveau one. The question is not equivalent to the Tate conjecture; it is a standalone structural question about torsion in Tate-twisted étale cohomology over algebraic closures of finite fields. Schreieder states it is open to the best of his knowledge and expects the equality fails in general; Remark 2.4 links it to the vanishing of the natural map $H^n(X,\mathbb{Z}_\ell(n))\to K^M_n(k(X))^{\wedge}\otimes\mathbb{Z}_\ell$: for a given variety $X$, a positive answer for all $r\geq1$ forces this map to be zero, so that the completed Milnor K-group then captures no information from $H^n(X,\mathbb{Z}_\ell(n))$. The source records only this one-way implication; the converse is not established there.

Problem Statement

Let $k=\bar{\mathbb{F}}_p$ be an algebraic closure of a finite field, let $X/k$ be a smooth projective variety, let $\ell\neq p$ be a prime, and let $n,r\geq1$ be integers. Let $N^1H^n$ denote the coniveau-one part of $H^n(X_{\mathrm{\acute{e}t}},\mu_{\ell^r}^{\otimes n})$, i.e. the subgroup of classes supported on a divisor: a class lies in $N^1H^n$ if it vanishes after restriction to the complement of some divisor of $X$, and equivalently $N^1H^n$ is generated by Gysin pushforwards (refined cycle classes) from divisors. Is the inclusion an equality,

$$H^n\!\left(X_{\mathrm{\acute{e}t}},\mu_{\ell^r}^{\otimes n}\right)=N^1H^n\!\left(X_{\mathrm{\acute{e}t}},\mu_{\ell^r}^{\otimes n}\right),$$

that is, is every Tate-twisted $\ell^r$-torsion class supported on a divisor? (As context recorded in Schreieder's Remark 2.4, a positive answer for all $r\geq1$ for a given $X$ forces the natural map

$$H^n\!\left(X,\mathbb{Z}_\ell(n)\right)\longrightarrow K^M_n(k(X))^{\wedge}\otimes\mathbb{Z}_\ell$$

to be zero, where $K^M_n(k(X))^{\wedge}\otimes\mathbb{Z}_\ell=\lim_r\, K^M_n(k(X))\otimes\mathbb{Z}/\ell^r$ denotes the $\ell$-adic completion of the Milnor K-theory of the function field $k(X)$. This vanishing is a consequence of the equality, not a formulation equivalent to it: the source records only this one-way implication and does not establish the converse, so proving the vanishing of the map does not by itself resolve the question.) The cases $n=1$ and $n=2$ are known and the equality fails there in general, so the open content is $n\geq3$: either prove the equality for every smooth projective $X$ over $\bar{\mathbb{F}}_p$, every prime $\ell\neq p$, every $n\geq3$ and every $r\geq1$, or construct an explicit counterexample — a specific $X$, $\ell$, $n\geq3$, $r$, and a class in $H^n(X,\mu_{\ell^r}^{\otimes n})$ that does not lie in $N^1H^n$, i.e. a nonzero class in unramified cohomology $H^n_{\mathrm{nr}}(X,\mu_{\ell^r}^{\otimes n})=H^n/N^1H^n$ (the subgroup of classes in $H^n(k(X),\mu_{\ell^r}^{\otimes n})$ whose residues at all codimension-one points of $X$ vanish).

References

1. Stefan Schreieder, 'Remarks on Milnor K-theory and Tate's conjecture for divisors' (2024), arXiv:2406.18706, DOI: 10.48550/arXiv.2406.18706, https://arxiv.org/abs/2406.18706
2. Spencer Bloch and Hélène Esnault, 'The coniveau filtration and non-divisibility for algebraic cycles', Mathematische Annalen 304 (1996), 303–314, DOI: 10.1007/BF01446296, https://doi.org/10.1007/BF01446296
3. Samet Balkan and Stefan Schreieder, 'Cycle conjectures and birational invariants over finite fields' (v1 June 2024, v2 December 2025), arXiv:2406.14438, DOI: 10.48550/arXiv.2406.14438, https://arxiv.org/abs/2406.14438
4. Jean-Louis Colliot-Thélène and Stefan Schreieder, 'Divisibility phenomena in motivic Bloch–Ogus theory' (2026), arXiv:2605.22494, DOI: 10.48550/arXiv.2605.22494, https://arxiv.org/abs/2605.22494
5. Jean-Louis Colliot-Thélène and Bruno Kahn, 'Cycles de codimension 2 et $H^3$ non ramifié pour les variétés sur les corps finis' (2011), arXiv:1104.3350, DOI: 10.48550/arXiv.1104.3350, https://arxiv.org/abs/1104.3350 (published in J. K-Theory 11 (2013), 1–53, DOI: 10.1017/is012009001jkt194)