Néron--Severi groups of proper schemes over finite fields

Abstract: Let (X) be a proper reduced scheme over a finite field (k), let (\ell) be a prime different from (\operatorname{char} k), and write (\ol X=X\times_{k}\ol k) for its base change to an algebraic closure (\ol k) of (k). Call a class in (\rH^{2}_{\et}(\ol X,\bbZ_{\ell}(1))) \emph{Zariski-locally trivial} if it vanishes on a Zariski-open cover of (\ol X). We prove that the first Chern class map identifies (\NS(\ol X)\otimes\bbZ_{\ell}) with the group of Zariski-locally trivial classes whose image in (\rH^{2}_{\et}(\ol X,\Ql(1))) has weight zero. This is the finite-field analogue of a theorem of Barbieri-Viale--Rosenschon--Srinivas for proper seminormal complex varieties. In the finite-field setting neither seminormality nor irreducibility is needed.

• The paper proves an ℓ-adic isomorphism characterizing the Néron–Severi group for proper reduced schemes over finite fields.
• It refines classical cycle class maps by linking Zariski-local triviality and weight-zero conditions under Frobenius action.
• The authors employ descent techniques and cohomological methods that bypass the need for seminormality and irreducibility assumptions.

Néron–Severi Groups of Proper Schemes over Finite Fields

Background and Motivation

The paper "Néron–Severi groups of proper schemes over finite fields" (2607.11777) addresses a fundamental question concerning the algebraic cycles of codimension one (divisors) on proper schemes over finite fields, examining their relationship with étale and Zariski cohomology. The Néron–Severi group $\mathrm{NS}(X)$ is the quotient $\mathrm{Pic}(X)/\mathrm{Pic}^0(X)$, encoding divisor classes modulo those algebraically equivalent to zero. Over $\mathbb{C}$, the Lefschetz theorem on $(1,1)$-classes and Hodge theory establish a tight relationship between $\mathrm{NS}(X)$ and certain subgroups of topological and algebraic cohomology. In singular or non-normal settings, these classical correspondences break down; however, prior work by Barbieri-Viale, Rosenschon, and Srinivas demonstrated that their intersection can be characterized for seminormal complex varieties. The current paper proves the precise finite-field analogue, with significant refinements: seminormality and irreducibility assumptions are not required.

Main Results

The central achievement is a characterization of the $\ell$-adic realization of the Néron–Severi group $\mathrm{NS}(\overline{X})\otimes\mathbb{Z}_\ell$, for proper reduced schemes $X$ over a finite field $k$, in terms of Zariski-locally trivial classes in $\mathrm{H}^2(\overline{X},\mathbb{Z}_\ell(1))$ whose images are pure of weight zero under the action of geometric Frobenius. The main theorem asserts:

$\mathrm{Pic}(X)/\mathrm{Pic}^0(X)$0

where $\mathrm{Pic}(X)/\mathrm{Pic}^0(X)$1 denotes the subgroup of Zariski-locally trivial classes, and the weight-zero condition corresponds to Frobenius eigenvalues of absolute value $\mathrm{Pic}(X)/\mathrm{Pic}^0(X)$2. This formally generalizes the classical Lefschetz property in characteristic zero and refines the divisor-$\mathrm{Pic}(X)/\mathrm{Pic}^0(X)$3-adic cycle class map as considered in the Tate conjecture.

Key Numerical Examples

The paper emphasizes that the cycle class map targets only the algebraic part of cohomology (Picard rank), which may be strictly smaller than the full cohomological dimension: for non-supersingular abelian surfaces, $\mathrm{Pic}(X)/\mathrm{Pic}^0(X)$4, and for finite-height K3 surfaces, $\mathrm{Pic}(X)/\mathrm{Pic}^0(X)$5, illustrating the necessity of both the Zariski-local triviality and weight-zero conditions.

Contradictory and Sharp Claims

Crucially, the authors demonstrate that neither seminormality nor irreducibility is necessary for the theorem over finite fields, contradicting known complex-analytic pathologies—over $\mathrm{Pic}(X)/\mathrm{Pic}^0(X)$6, these conditions are essential ("cannot be dropped"). Furthermore, they provide explicit examples (e.g., normal projective surfaces with extra weight $\mathrm{Pic}(X)/\mathrm{Pic}^0(X)$7 classes, as in Example~1 of Barbieri-Viale and Srinivas) that show the weight-zero restriction is genuinely sharp: Zariski-local triviality alone does not suffice.

Technical Approach

Sheaf-Theoretic and Cohomological Methods

The paper employs a comparison between étale, pro-étale, and Zariski cohomology, leveraging the Bloch–Ogus coniveau spectral sequence, proper and sdh-topology hypercovers (inspired by de Jong alterations), and advanced descent mechanisms (e.g., Bhatt–Scholze $\mathrm{Pic}(X)/\mathrm{Pic}^0(X)$8-descent for vector bundles). The key object in the analysis is the Zariski sheaf $\mathrm{Pic}(X)/\mathrm{Pic}^0(X)$9, which provides a bridge between line bundles and their cycle classes.

A pivotal step is the reduction to seminormal schemes via universal homeomorphisms and sdh-sheafification. This allows for hypercover descent from smooth proper varieties, even in positive characteristic, employing spectral sequence arguments to separate weights under Frobenius action.

Frobenius Module Theory

Building on Deligne's theory of weights, the authors carefully analyze the structure of cohomology groups as Frobenius modules, extracting direct sum decompositions and leveraging exactness properties of weight gradings. The resulting technical apparatus ensures the weight-zero filtration is both necessary and sufficient, and is robust under descent (hypercover and seminormalization).

Cycle Class and Tate Conjecture

The cycle class map is interpreted via the Kummer sequence, and further compared to the finite-order part of étale cohomology, as predicted by the divisor case of the Tate conjecture. Assuming the Tate conjecture for divisors, the result extends to all proper reduced schemes—not just smooth proper ones—yielding a three-way identification among the Néron–Severi group, Zariski-local weight-zero cohomology, and the finite-order part.

Implications and Future Directions

Theoretical Significance

This work establishes a transparent, weight-theoretic, and functorial dictionary between divisors and their cohomological avatars for all proper reduced schemes over finite fields. It clarifies the limitations and sharpness of classical analogues from complex geometry and serves as a robust foundation for further studies in algebraic cycles, motivic cohomology, and the formulation and testing of generalizations of the Tate conjecture.

Practical Considerations

The techniques employed, notably descent via sdh-hypercovers and Frobenius weight separation, are of broad utility in the study of cohomology and cycle theory in positive characteristic. The results ensure that, for tasks involving the computation of divisor classes or the verification of cycle-theoretic conjectures, the practitioner can confidently dispense with assumptions of seminormality or irreducibility, provided the relevant weights and Zariski localization are maintained.

Anticipated Developments

Further advancements may integrate these findings with the theory of motives, pursue generalizations to higher codimension cycles, or refine the understanding of the relationship between $\mathbb{C}$0-adic cohomological invariants and algebraic cycles on singular, non-reduced, or geometrically intricate schemes. The explicit connection to the finite-order part and applications to the Tate conjecture signal potential breakthroughs in the classification and computation of rational and integral cycles over finite fields.

Conclusion

The paper provides a rigorous characterization of $\mathbb{C}$1-adic cycle classes arising from divisors for proper reduced schemes over finite fields, identifying the Néron–Severi group with Zariski-locally trivial weight-zero cohomology classes. This refines classical analogues, removes restrictive assumptions, and situates the problem within the broader context of the Tate conjecture, spectral sequence descent, and Frobenius module theory. The methodology and results will serve as a foundational reference for subsequent research in the arithmetic and cohomological study of algebraic cycles in characteristic $\mathbb{C}$2 (2607.11777).