Lichtenbaum's finiteness conjecture for integral étale motivic cohomology of arithmetic schemes

For an arithmetic scheme $X$ — a regular scheme proper over the spectrum of the ring of integers of a number field, or over a smooth proper curve over a finite field — the integral étale motivic cohomology groups $H^i_{\mathrm{et}}(X,\mathbb{Z}(n))$ are conjectured to be finitely generated for $i\le 2n$, finite for $i=2n+1$, and of cofinite type for $i\ge 2n+2$. This is Lichtenbaum's finiteness conjecture, formulated as Conjecture 3.1 in Geisser's "Duality for integral motivic cohomology" (arXiv:1712.09021), where it is the load-bearing open hypothesis for the expected integral duality pairings and for special-value formulas for zeta functions of arithmetic schemes. Over finite fields, finiteness in degree $2n+1$ is equivalent to Tate's conjecture on surjectivity of the cycle class map, so the conjecture subsumes one of the central open problems of arithmetic geometry. The conjecture is assumed throughout the Weil–étale literature (by Lichtenbaum, Flach–Morin, and Beshenov) and is not known to be proved or refuted; the problem remains open.

Background

Motivic cohomology is the cohomology theory of algebraic cycles that underlies and refines algebraic $K$-theory. For a scheme $X$ it is defined as the hypercohomology of Bloch's motivic complex $\mathbb{Z}(n)$, a chain complex built from algebraic cycles of codimension $n$ on the products of $X$ with algebraic simplices; its homological shadow, the higher Chow groups $CH^n(X,m)$, extends the classical Chow group of codimension-$n$ cycles modulo rational equivalence to higher degrees $m$. Taking hypercohomology on the étale site (the topology whose covers are finite étale maps) gives étale motivic cohomology $H^i_{\mathrm{et}}(X,\mathbb{Z}(n))$, an integral, torsion-sensitive avatar of $K$-theory: via the motivic-to-$K$-theory spectral sequence the groups with $i\le 2n$ are the building blocks of $K_{2n-i}(X)$, and after inverting or completing at primes they are computed by étale cohomology through the now-proved Bloch–Kato and Lichtenbaum–Quillen comparisons.

An arithmetic scheme in this problem means a regular scheme $X$ proper over a base $B$ which is either the spectrum of the ring of integers $\mathcal{O}_K$ of a number field, or a smooth proper curve over a finite field. For such $X$ the groups $H^i_{\mathrm{et}}(X,\mathbb{Z}(n))$ are expected to have a rigid integral structure: finitely generated in degrees $i\le 2n$, finite in degree $i=2n+1$, and of cofinite type — a direct sum of a finite group and finitely many copies of $\mathbb{Q}/\mathbb{Z}$ — in degrees $i\ge 2n+2$. The expectation goes back to Lichtenbaum's vision that special values of the zeta function of an arithmetic scheme should be expressed as orders and Euler characteristics of these groups, developed in his papers on the Weil–étale topology, a refinement of the étale topology of arithmetic schemes in which the Galois group of a finite-field residue is replaced by a Weil group so that integral cohomology behaves like the cohomology of a compact space, with locally compact groups and Pontrjagin duality between a discrete abelian group $A$ and its dual $\operatorname{Hom}(A,\mathbb{Q}/\mathbb{Z})$. Geisser, building on his computation of Weil–étale cohomology over finite fields and his construction of duality pairings via cycle complexes, states the finiteness expectation as Conjecture 3.1 of arXiv:1712.09021 and proves that, jointly with the existence of certain pairings, it yields perfect duality pairings between the completion of $H^j_{\mathrm{et}}(X,\mathbb{Z}(u))$ and the torsion subgroup of $H^{i+1}_c(X,\mathbb{Z}(n))$, with values in $\mathbb{Q}/\mathbb{Z}$, where $H^{\bullet}_c$ denotes cohomology with compact support along the base.

The conjecture is known only in the simplest cases, for example when $X$ is the spectrum of a ring of integers itself, where it reduces to classical finite-generation and cofiniteness theorems for the $K$-theory and étale cohomology of number rings; the open content lies in higher relative dimension. Two facts shape the difficulty. First, over finite fields Geisser proves (Proposition 3.2) that finiteness of $H^{2n+1}_{\mathrm{et}}(X,\mathbb{Z}(n))$ is equivalent to Tate's conjecture in degree $n$, the assertion that for every prime $\ell$ the cycle class map from codimension-$n$ cycles to the Galois-invariant $\ell$-adic étale cohomology $H^{2n}_{\mathrm{et}}(\bar{X},\mathbb{Z}_\ell(n))$ becomes surjective after tensoring with $\mathbb{Z}_\ell$; hence the finiteness conjecture subsumes one of the standard open problems of the field. Second, for $X$ regular but not smooth over $B$ (schemes with bad reduction) the basic cycle-level tools are missing: the cup product on higher Chow groups with compact support is not constructed in general — Spitzweck announced a construction for $X$ smooth over the base via his commutative motivic Eilenberg–Mac Lane spectrum over Dedekind domains — and the required finite-coefficient duality is known with $\mathbb{Z}/m(n)$ replaced by Sato's $p$-adic Tate twists $\mathfrak{T}_m(n)$ (modified étale sheaf complexes encoding crystalline-type information at primes of bad reduction) only when the fibers over places dividing $m$ are normal crossing schemes, with Zhong's comparison of dualizing complexes giving partial results toward identifying the two complexes. The finiteness conjecture itself is independent of these pairing ingredients and stands as the central open finiteness statement for integral motivic cohomology in the arithmetic setting.

Problem Statement

Let $B$ be either $\operatorname{Spec}\mathcal{O}_K$, the spectrum of the ring of integers of a number field $K$, or a smooth proper curve over a finite field, and let $X$ be a regular scheme that is proper over $B$. For integers $n\ge 0$ and any $i$, write $H^i_{\mathrm{et}}(X,\mathbb{Z}(n))$ for the étale hypercohomology of Bloch's motivic complex $\mathbb{Z}(n)$. Prove, or refute, Lichtenbaum's finiteness conjecture (Conjecture 3.1 of Geisser, arXiv:1712.09021): $H^i_{\mathrm{et}}(X,\mathbb{Z}(n))$ is a finitely generated abelian group for $i\le 2n$, a finite group for $i=2n+1$, and of cofinite type — that is, isomorphic to the direct sum of a finite group and finitely many copies of $\mathbb{Q}/\mathbb{Z}$ — for $i\ge 2n+2$. The statement is required for all schemes $X$ in the class above, all $n\ge 0$, and all degrees $i$; a refutation consists of an explicit $X$, $n$, and $i$ together with a rigorous computation of $H^i_{\mathrm{et}}(X,\mathbb{Z}(n))$ violating the asserted structure in that degree. The companion open ingredients of the source paper — the cup-product pairing with compact support on higher Chow groups and the finite-coefficient duality over arithmetic bases — are context for the duality program, not additional required deliverables of this problem.

References

1. T. H. Geisser, Duality for integral motivic cohomology, arXiv:1712.09021 (2017), DOI 10.48550/arXiv.1712.09021, https://arxiv.org/abs/1712.09021 (source paper)
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