Finite generation of motivic, Frobenius, Kato, and Weil-étale (co)homology of schemes over finite fields (Geisser's Conjecture 2.5 / Lichtenbaum's finiteness)

For a separated scheme $X$ of finite type over a finite field, Geisser constructed two intermediate homology theories between motivic cohomology and Weil-étale cohomology — Frobenius homology $H_i^F(X,\mathbb{Z}(n))$ and Kato homology $H_i^K(X,\mathbb{Z}(n))$ — and conjectured (Conjecture 2.5 of arXiv:1103.5544, published in Contemp. Math. 571) that these, together with motivic Borel–Moore homology $H_i^c(X,\mathbb{Z}(n))$, are finitely generated abelian groups for every $i$ and $n\ge 0$. For smooth proper $X$ the source states the companion Conjecture 3.3, which additionally asks finite generation of Weil-étale cohomology; that Weil-étale component is equivalent to Lichtenbaum's structure conjecture for étale motivic cohomology and to the strong form of Tate's and Beilinson's conjectures, and implies Parshin's conjecture. Despite sustained work on the component conjectures (Tate, Parshin, Brauer finiteness) through 2026, the finite generation conjectures remain open in general, and this record asks for a proof or refutation of them.

Background

Algebraic cycles and motivic cohomology are the standard tools for studying integer-valued invariants of algebraic varieties. For a smooth variety $X$, Bloch's cycle complex $\mathbb{Z}(n)$ is the complex of algebraic cycles of codimension $n$ meeting faces of simplices properly; its Zariski hypercohomology $H^i_{\mathcal{M}}(X,\mathbb{Z}(n))$ is motivic cohomology, which for smooth $X$ agrees with the higher Chow groups $CH^{n}(X,2n-i)$ and, after tensoring with $\mathbb{Q}$, with the graded pieces of the gamma filtration on algebraic $K$-theory. The variant $\mathbb{Z}^{c}(n)$, built from cycles of dimension $n$ (homological indexing), defines motivic Borel–Moore homology $H_i^{c}(X,\mathbb{Z}(n))=H_i(\mathbb{Z}^{c}(n)(X))$, a theory with functoriality and long exact localization sequences for arbitrary finite-type schemes, with or without smoothness.

Over a finite field $\mathbb{F}_q$ with algebraic closure $\bar{\mathbb{F}}_q$, let $\varphi$ be the geometric Frobenius and $G=\langle\varphi\rangle$ the Weil group of $\mathbb{F}_q$, the cyclic group generated by $\varphi$. One can form arithmetic homology theories from the action of $\varphi$ on cycles over $\bar{X}=X\otimes\bar{\mathbb{F}}_q$: Frobenius homology $H_i^{F}(X,\mathbb{Z}(n))$ is the homology of the double complex $\mathbb{Z}^{c}(n)(\bar{X})\xrightarrow{\varphi-1}\mathbb{Z}^{c}(n)(\bar{X})$, and Kato homology $H_i^{K}(X,\mathbb{Z}(n))$ is the homology of the complex of coinvariants $(\mathbb{Z}^{c}(n)(\bar{X}))_{G}$; the latter generalizes the integral Kato homology of arithmetic schemes, which is the case $n=\dim X$. Lichtenbaum's Weil-étale cohomology $H^i_{W}(X,\mathbb{Z}(n))$ is the cohomology of $R\Gamma_{G}R\Gamma_{\mathrm{et}}(\bar{X},\mathbb{Z}(n))$, where $\mathbb{Z}(n)$ is now viewed as a complex of étale sheaves; it is designed to capture special values of zeta functions integrally. For smooth proper $X$ these theories fit into a commutative diagram with exact rows, $H^i_{\mathcal{M}}(X,\mathbb{Z}(n))\to H^i_F(X,\mathbb{Z}(n))\to H^{i-1}_K(X,\mathbb{Z}(n))$ over $H^i_{\mathrm{et}}(X,\mathbb{Z}(n))\to H^i_W(X,\mathbb{Z}(n))\to H^{i-1}_{\mathcal{M}}(X,\mathbb{Q}(n))$, so their finite generation can be studied comparatively.

The finite-generation questions generalize classical conjectures. Bass conjectured that the higher $K$-groups of a regular scheme of finite type over $\mathbb{Z}$ are finitely generated, which over $\mathbb{F}_q$ becomes finite generation of motivic cohomology; Tate's conjecture predicts that for smooth projective $X$ the cycle map $CH^n(X)\otimes\mathbb{Q}_l\to H^{2n}_{\mathrm{et}}(\bar{X},\mathbb{Q}_l(n))^{G}$ is surjective; Beilinson's conjecture predicts its rational injectivity; and Parshin's conjecture $P_n$ predicts that $CH_n(X,i)$ is torsion for $i\ne 0$ on smooth projective $X$ (equivalently, rational motivic cohomology vanishes outside bidegree $(2n,n)$). Work cited in the source paper shows that finite generation of Weil-étale cohomology for smooth proper $X$ is equivalent to the strong form of Tate's and Beilinson's conjectures, and the source paper proves (its Proposition 5.1) that Lichtenbaum's structure conjecture for $H^i_{\mathrm{et}}(X,\mathbb{Z}(n))$ — finite outside degrees $2n$ and $2n+2$, finitely generated in degree $2n$, of cofinite type in degree $2n+2$ — is equivalent to finite generation of Weil-étale cohomology and implies Parshin's conjecture.

Known low-weight cases (from the source paper and later work): the conjectures hold for $n=0$ trivially, and for $n=1$ all groups in the diagram are finitely generated except possibly the Brauer group $\mathrm{Br}(X)$ and $H^3_W(X,\mathbb{Z}(1))$, whose finiteness is equivalent to Tate's conjecture for divisors; for $n=\dim X$ the conjecture is equivalent to Parshin's conjecture $P_0$ on zero-cycles; and the torsion-coefficient finiteness holds for $n=0$ (prime to the characteristic, and in general assuming resolution of singularities). Pirutka constructed, for infinitely many primes $p$, a smooth projective geometrically rational variety of dimension 5 over $\mathbb{F}_p$ with a nonzero class in unramified cohomology $H^3_{\mathrm{nr}}(X,\mathbb{Z}/2)$ — the subgroup of the Galois cohomology $H^3(F(X),\mathbb{Z}/2)$ of the function field $F(X)$ consisting of classes unramified at every codimension-one point of $X$ — giving a nontrivial 2-torsion element of Kato homology $H^3_K(X,\mathbb{Z}(2))$, so the conjectural groups are genuinely interesting invariants rather than trivially zero. In the other direction, examples over algebraically closed fields of characteristic 0 (discussed in §4 of the source paper) show that the analogue of motivic finite generation over $\bar{X}$ fails in characteristic 0, while no characteristic-$p$ counterexample is known; this marks the boundary of what is expected.

Problem Statement

Are the groups $H_i^{c}(X,\mathbb{Z}(n))$ (motivic Borel–Moore homology), $H_i^{F}(X,\mathbb{Z}(n))$ (Frobenius homology), and $H_i^{K}(X,\mathbb{Z}(n))$ (Kato homology) finitely generated for every separated scheme $X$ of finite type over a finite field $\mathbb{F}_q$, every integer $i$, and every $n\ge 0$? This is Conjecture 2.5 of Geisser's 'Finite generation conjectures for cohomology over finite fields'. The source notes that, by localization, it suffices to consider smooth proper $X$ (granting that every finitely generated field over $\mathbb{F}_q$ has a smooth proper model); there it states the companion Conjecture 3.3, which asks that the bold terms of its comparison diagram — motivic cohomology $H^i_{\mathcal{M}}(X,\mathbb{Z}(n))$, Frobenius cohomology $H^i_{F}(X,\mathbb{Z}(n))$, Kato cohomology $H^{i-1}_{K}(X,\mathbb{Z}(n))$, and Weil-étale cohomology $H^i_{W}(X,\mathbb{Z}(n))$ — be finitely generated for all $i$ and $n$. The Weil-étale component alone is equivalent to Lichtenbaum's structure conjecture (Conjecture 3.4): $H^i_{\mathrm{et}}(X,\mathbb{Z}(n))$ is finite for $i\notin\{2n,2n+2\}$, finitely generated for $i=2n$, and of cofinite type for $i=2n+2$; by earlier work of the author it is also equivalent to the strong form of Tate's and Beilinson's conjectures, and by Proposition 5.1 of the source it implies Parshin's conjecture. The weaker companion with finite coefficients (Conjecture 2.9) asks that all terms in the exact sequence $\cdots\to H_i^{c}(X,\mathbb{Z}/m(n))\to H_{i+1}^{F}(X,\mathbb{Z}/m(n))\to H_{i+1}^{K}(X,\mathbb{Z}/m(n))\to\cdots$ be finite for all $i$, $n$, $X$, and $m$. Note that an affirmative answer to the companion Conjecture 3.3 (or its Weil-étale component) resolves Tate's, Beilinson's, Lichtenbaum's, and Parshin's conjectures for smooth proper $X$, that Conjecture 2.5 itself implies the torsion finiteness of Conjecture 2.9, and that a negative answer exhibits a specific $X$, $i$, $n$ with one of the three groups not finitely generated.

References

1. Thomas H. Geisser, Finite generation conjectures for motivic cohomology theories over finite fields, Contemporary Mathematics 571 (2012), 153–165, DOI 10.1090/conm/571/11326; arXiv:1103.5544, https://arxiv.org/abs/1103.5544
2. Thomas H. Geisser, On the structure of étale motivic cohomology, Journal of Pure and Applied Algebra 221 (2017), 1614–1628, DOI 10.1016/j.jpaa.2016.12.019; arXiv:1609.02273, https://arxiv.org/abs/1609.02273
3. Thomas H. Geisser, Brauer and Néron–Severi groups of surfaces over finite fields, Journal of Number Theory 284 (2026), 38–61, DOI 10.1016/j.jnt.2025.12.004; arXiv:2401.03709, https://arxiv.org/abs/2401.03709
4. Thomas H. Geisser, Tate's conjecture and the Tate–Shafarevich group over global function fields, Journal of the Institute of Mathematics of Jussieu 20 (2021), no. 3, 1001–1022, DOI 10.1017/S147474801900046X; arXiv:1801.02406, https://arxiv.org/abs/1801.02406
5. Thomas Geisser, Parshin's conjecture and motivic cohomology with compact support, arXiv:1002.0105 (2010), https://arxiv.org/abs/1002.0105
6. Oliver Gregory, Motivic cohomology and K-theory of some surfaces over finite fields, Journal of Pure and Applied Algebra 228 (2024), article 107518, DOI 10.1016/j.jpaa.2023.107518; arXiv:2302.09867, https://arxiv.org/abs/2302.09867
7. Alena Pirutka, Sur le groupe de Chow de codimension deux des variétés sur les corps finis, Algebra & Number Theory 5 (2011), 803–817, DOI 10.2140/ant.2011.5.803, https://doi.org/10.2140/ant.2011.5.803
8. Aydin Yousefzadehfard, Proof of the Parshin's Conjecture, arXiv:2011.05544 (2020), https://arxiv.org/abs/2011.05544
9. Bruno Kahn, An approach to the Tate conjecture for surfaces over a finite field, arXiv:2505.07337 (2025), https://arxiv.org/abs/2505.07337
10. Toshiro Hiranouchi and Rin Sugiyama, Zero-cycles on varieties over a $\mathfrak{B}_s$-field, arXiv:2509.15617 (2025, last revised 2026), https://arxiv.org/abs/2509.15617