Finite Generation of Cohomology

• Finite generation of cohomology is the property that a cohomology ring or module is finitely generated, enabling structured analysis of algebraic cycles and arithmetic invariants.
• It interlinks multiple cohomology theories such as motivic, Frobenius, and Weil-étale using exact sequences and spectral sequences to relate algebraic and geometric properties.
• Advances in finite generation provide pivotal tools for validating conjectures like Bass and Parshin, linking cycle maps to arithmetic dualities.

Finite generation of cohomology refers to the property that a given (usually graded) cohomology ring or module is finitely generated as an algebra or as a module over such an algebra. In algebraic geometry, number theory, and representation theory, finite generation serves as a critical structural property enabling the application of geometric invariants, support varieties, and deep connections with major conjectures and phenomena. The property is often conjectural in general, but has been established for significant classes of objects, especially over finite fields, finite groups, finite group schemes, and certain types of (braided) Hopf algebras.

1. Fundamental Cohomology Theories and Finite Generation Problems

Several central cohomology theories arise in the paper of algebraic varieties and schemes over finite fields: motivic cohomology $H_M(X, \mathbb{Z}(n))$, Weil-étale cohomology $H_W(X, \mathbb{Z}(n))$, and more recently, intermediate (or Frobenius) cohomology $H_F(X, \mathbb{Z}(n))$. For smooth varieties $X$ over a finite field, motivic cohomology is related to higher Chow groups via $H_M(X, \mathbb{Z}(n)) \cong CH_n(X, *)$, while Weil-étale cohomology is intertwined with special values of zeta functions and conjectures of Tate and Beilinson. The intermediate cohomology $H_F(X, \mathbb{Z}(n))$ is constructed by explicitly incorporating the Frobenius action into Bloch's cycle complex, producing a complex whose homology encodes arithmetic information specific to the finite field setting (Geisser, 2011).

The finite generation question for these cohomology groups lies at the intersection of several major conjectures:

• Bass conjecture (for motivic cohomology): finite generation of higher algebraic $K$-groups for schemes of finite type.
• Beilinson–Tate conjecture (for Weil-étale cohomology): equivalence of finite generation of $H_W(X, \mathbb{Z}(n))$ with strong forms of Tate and Beilinson conjectures.
• Frobenius cohomology serves as an intermediary: its conjectured finite generation provides a bridge, allowing reduction or comparison of finite generation problems across motivic and Weil-étale settings.

Natural exact and long exact sequences connect these cohomology theories, with Kato cohomology $H_K(X, \mathbb{Z}(n))$ measuring the "defect" or discrepancy between algebraic (motivic) and geometric (étale or Weil-étale) phenomena.

2. Construction of Intermediate (Frobenius) Cohomology and Key Exact Sequences

The construction of $H_F(X, \mathbb{Z}(n))$ begins from Bloch's cycle complex $Z(n)$. The motivic cohomology is given by

$$H_M(X, \mathbb{Z}(n)) = H^{2d+1-\bullet}\big(X, \mathbb{Z}(d-n)\big)$$

for $X$ smooth of pure dimension $d$. The Frobenius action is imposed by considering a double complex

$Z^\prime(n)(X) \to Z^c(n)(X)$

with geometric Frobenius $y$ acting covariantly, and one obtains

$H_F(X, \mathbb{Z}(n)) = H_*(R\Gamma_G Z(n)(X)).$

In this framework, the following comparison diagram organizes the relationships: $$\begin{array}{cccccc} H_M(X, \mathbb{Z}(n)) & \to & H_F(X, \mathbb{Z}(n)) & \to & H_K(X, \mathbb{Z}(n)) & \to \cdots \ & & \downarrow & & & \ & & H_{et}(X, \mathbb{Z}(n)) & \to & H_W(X, \mathbb{Z}(n)). & \end{array}$$ Exact sequences such as

$$0 \to H_E(X, \mathbb{Z}(n))^G \to H_E(X, \mathbb{Z}(n)) \to H_{E_1}(X, \mathbb{Z}(n))^G \to 0$$

and long exact sequences relating motivic, Kato, and Weil-étale invariants quantify the obstruction to lifting finite generation from motivic to Weil-étale cohomology (Geisser, 2011).

3. Relationship to Classical and Modern Conjectures

The finite generation of motivic cohomology for smooth projective varieties over finite fields is classically equivalent (via the Bloch–Lichtenbaum spectral sequence) to the Bass conjecture. For Weil-étale cohomology, finite generation is tightly linked to the strongest forms of the Tate and Beilinson conjectures: for $X$ smooth and proper, $H_W(X, \mathbb{Z}(n))$ is finitely generated if and only if rational motivic and étale cohomology agree in degree $2n$ (and the relevant "error terms" in degree $2n+1$ vanish).

The paper proves that, under Lichtenbaum’s conjecture for étale cohomology, finite generation of $H_W(X, \mathbb{Z}(n))$ implies Parshin’s conjecture, and that the Frobenius (intermediate) cohomology serves as a device to interpolate between motivic and Weil-étale finite generation statements. The precise dependencies of conjectures can be visualized as: $$\text{Lichtenbaum’s conjecture} \ \Longrightarrow\ \text{Finite generation of}\ H_W(X,\mathbb{Z}(n)) \ \Longrightarrow\ \text{Parshin’s conjecture}$$

The "defect" $H_K(X, \mathbb{Z}(n))$ is interpreted as quantifying the failure of the integral Tate conjecture in the sense that its vanishing corresponds to the cycle map being an isomorphism.

4. Spectral Sequences, Niveau Filtration, and Local-Global Analysis

Spectral sequence methods underlie much of the analysis concerning finite generation. One key feature is the niveau filtration spectral sequence, in which the $E_1$-term is built from Chow groups over residue fields: $$E^1_{s,t} = CH_s(k(x), A(s-n)) \Longrightarrow H_{s+t}(X, A(n)),$$ where the sum runs over $x \in X_{(s)}$ (points of dimension $s$). This decomposition reveals how finite generation in global cohomology depends on the structure of local Chow groups and their cycles, as well as their behavior under Frobenius.

Frobenius cohomology fits naturally into these spectral sequences, and the permanence of certain cycles is integral for establishing Noetherian properties and thereby finite generation of the abutment.

The introduction of Kato cohomology further allows "localization" of obstructions: Kato cohomology can be interpreted as unramified (or coinvariant) cycles, providing a finer bridge between algebraic and topological invariants, often via explicit coinvariant complexes and their exactness properties.

5. Comparison of Cohomology Theories and Arithmetic Implications

Comparison of motivic, Frobenius, and Weil-étale cohomology is facilitated by natural maps and exact sequences. These relationships measure, for instance, the failure of motivic classes to be detected by Galois actions (étale or Weil-étale realization).

A prominent formula highlighted is: $H_K(X, \mathbb{Z}(n)) \cong H_{* - 2n}(KC_n(X)),$ where $KC_n(X)$ denotes a Kato-type complex. The vanishing of $H_K$ outside a narrow range indicates tight control over "defect classes," corresponding to failure of the cycle map and providing evidence that the intermediate cohomology is a central invariant in the arithmetic of $X$ over finite fields.

In practice, these connections allow one to track arithmetic invariants such as special values of zeta functions, the structure of Galois modules, and the interplay of algebraic cycles with arithmetic duality.

6. Technical Advances, Exact Sequences, and Structural Insights

The paper introduces several novel technical tools:

• Construction of a double complex incorporating the Frobenius action directly into the cycle complex.
• Systematic analysis of Kato cohomology via coinvariants under the Weil group.
• Spectral sequence structures illustrating the assembly of global cohomology from local Chow group data.
• Precise exact sequences comparing kernels and cokernels of cycle maps and "change of topology" maps, which allow for the translation of integral phenomena into $\ell$-adic or Weil-étale frameworks.

Key exact sequences include those relating the integral structure of motivic cohomology with its étale and Weil-étale analogs—critical for analyzing the detectability of cycles and for establishing finite generation in the presence of torsion.

These techniques not only clarify the conceptual framework for understanding finite generation but also supply detailed tools for ongoing research into algebraic cycles, arithmetic of schemes over finite fields, and the deeper structure of motivic and Weil-étale cohomology.

7. Synthesis and Broader Implications

The paper of finite generation of cohomology in the context of motivic/Frobenius/Weil-étale theories over finite fields demonstrates a deep synergy between algebraic cycles, homotopical methods, and arithmetic duality. The introduction of intermediate cohomology and the explicit spectral sequence and exact sequence machinery allow for a structural understanding of the relationships between different cohomology theories and the conjectures that govern them.

These advances establish that finite generation phenomena should be understood not as isolated features of distinct cohomology theories, but as manifestations of a single, interwoven arithmetic-extrasynthetic structure. The methodologies developed—Frobenius action, coinvariants, and spectral sequences—are now pivotal in broader questions regarding special values of $L$-functions, arithmetic duality, and the classification of invariants associated to varieties over finite fields.

Future work is anticipated to build further upon these exact sequences and spectral techniques, particularly as new types of intermediate or relative cohomology theories are investigated for their finite generation properties and arithmetic significance.