[Paper Review] Arithmetic homology and an integral version of Katos conjecture

Thomas Geisser|ArXiv.org|Apr 10, 2007

Geometry and complex manifolds7 citations

TL;DR

This paper introduces arithmetic homology, an integral Borel-Moore homology theory over finite fields, and an integral version of Kato homology. Using the Weil-etale formalism applied to Bloch's cycle complex, it establishes a long exact sequence relating higher Chow groups, arithmetic homology, and integral Kato homology, proving that the integral Kato conjecture holds for curves and providing a framework for understanding rational and torsion cycles on schemes over finite fields.

ABSTRACT

We define an integral Borel-Moore homology theory over finite fields, called arithmetic homology, and an integral version of Kato homology. Both types of groups are expected to be finitely generated, and sit in a long exact sequence with higher Chow groups of zero-cycles.

Motivation & Objective

• To define an integral Borel-Moore homology theory (arithmetic homology) over finite fields as a substitute for pathological etale higher Chow groups.
• To construct an integral version of Kato homology that measures the difference between higher Chow groups and arithmetic homology.
• To formulate and investigate an integral analog of Kato's conjecture for smooth, proper, and connected schemes over finite fields.
• To establish a long exact sequence connecting higher Chow groups, arithmetic homology, and integral Kato homology under standard conjectures.
• To prove the integral Kato conjecture in degree 0 and for curves, and to verify the long exact sequence in low degrees unconditionally.

Proposed method

• Arithmetic homology is defined as the cohomology of the complex $ R ilde{ au}_G R ilde{ au}(ar{X}_{ ext{et}}, \mathbb{Z}^c(0))[1] $, where $ G $ is the Weil group acting on $ \bar{X} = X \times_{\mathbb{F}_q} \bar{\mathbb{F}}_q $.
• The construction uses the Weil-etale formalism applied to Bloch's cycle complex $ \mathbb{Z}^c(0) $, yielding groups $ H_i^c(X_{\text{ar}}, \mathbb{Z}) $ expected to be finitely generated.
• Integral Kato homology $ H_i^K(X, \mathbb{Z}) $ is defined as the homology of a complex involving Milnor K-groups of residue fields tensored with $ \bar{\mathbb{F}}_q $, with Frobenius coinvariants.
• The long exact sequence is derived from the Beilinson-Lichtenbaum conjecture and the structure of localization sequences in etale cohomology and Weil-etale cohomology.
• The projective and homotopy bundle formulas for arithmetic homology are established via localization and the homotopy formula.
• The conjectural long exact sequence is shown to hold unconditionally in degrees $ i \leq 1 $, and the integral Kato conjecture is verified for curves.

Experimental results

Research questions

• RQ1 Does an integral Borel-Moore homology theory exist over finite fields that replaces the ill-behaved etale higher Chow groups?
• RQ2 Can an integral version of Kato homology be defined such that it measures the difference between higher Chow groups and arithmetic homology?
• RQ3 Is there a long exact sequence relating $ CH_0(X,i) $, $ H_{i+1}^c(X_{\text{ar}}, \mathbb{Z}) $, and $ H_{i+1}^K(X, \mathbb{Z}) $ for schemes over $ \mathbb{F}_q $?
• RQ4 Does the integral Kato conjecture hold for smooth, proper, and connected schemes over finite fields, with $ H_0^K(X, \mathbb{Z}) \cong \mathbb{Z} $ and $ H_i^K(X, \mathbb{Z}) = 0 $ for $ i > 0 $?
• RQ5 Can the long exact sequence be established unconditionally in low degrees, and is the conjecture verified for curves?

Key findings

• The integral Kato conjecture holds in degree 0 for all smooth, proper, and connected schemes over $ \mathbb{F}_q $, with $ H_0^K(X, \mathbb{Z}) \cong \mathbb{Z} $.
• For smooth and proper curves $ C $, the conjecture holds and the long exact sequence reduces to $ 0 \to (k(C) \otimes_{\mathbb{F}_q} \bar{\mathbb{F}}_q)^\times_G \to \bigoplus_{x \in C_{(0)}} \mathbb{Z} \to \mathbb{Z} \to 0 $.
• The long exact sequence relating $ CH_0(X,i) $, $ H_{i+1}^c(X_{\text{ar}}, \mathbb{Z}) $, and $ H_{i+1}^K(X, \mathbb{Z}) $ holds unconditionally for $ i \leq 1 $.
• Under the Beilinson-Lichtenbaum conjecture and resolution of singularities, the equivalence of three statements is proven: vanishing of rational Chow groups, the integral Kato conjecture, and the existence of the long exact sequence.
• The integral Kato homology groups $ H_i^K(X, \mathbb{Z}) $ are shown to be isomorphic to $ CH_0(X,i)_{\mathbb{Q}} $ for $ i = 1,2 $, when $ X $ is smooth and proper.
• The projective bundle and homotopy formulas for arithmetic homology are established, showing compatibility with standard motivic properties.