[Paper Review] 0-cycles with modulus on surfaces

Amalendu Krishna|arXiv (Cornell University)|Apr 13, 2015

Algebraic Geometry and Number Theory17 references3 citations

TL;DR

This paper establishes an exact sequence linking the Chow group with modulus $CH_0(X,D)$ and the relative $K$-group $K_0(X,D)$ for a smooth surface $X$ and effective Cartier divisor $D$. It resolves a question of Kerz and Saito for resolutions of singularities of normal surfaces and demonstrates that the localization sequence for ordinary Chow groups fails to extend to Chow groups with modulus.

ABSTRACT

Given a smooth surface $X$ over a field and an effective Cartier divisor $D$, we provide an exact sequence connecting $CH_0(X,D)$ and the relative $K$-group $K_0(X,D)$. We use this exact sequence to answer a question of Kerz and Saito whenever $X$ is a resolution of singularities of a normal surface. This exact sequence is used to show that the localization sequence for ordinary Chow groups does not extend to Chow groups with modulus.

Motivation & Objective

• To construct an exact sequence connecting $CH_0(X,D)$ and $K_0(X,D)$ for a smooth surface $X$ and effective Cartier divisor $D$.
• To answer a question posed by Kerz and Saito concerning the structure of $CH_0(X,D)$ when $X$ is a resolution of singularities of a normal surface.
• To investigate the failure of the localization sequence for ordinary Chow groups in the context of Chow groups with modulus.
• To clarify the structural differences between ordinary Chow groups and Chow groups with modulus on surfaces.

Proposed method

• Utilizes algebraic $K$-theory techniques to analyze the relative $K$-group $K_0(X,D)$ associated with the pair $(X,D)$.
• Applies the Gersten resolution and spectral sequence methods to relate $K_0(X,D)$ to the Chow group with modulus $CH_0(X,D)$.
• Employs the structure of the relative $K$-group to derive an exact sequence involving $CH_0(X,D)$ and $K_0(X,D)$.
• Analyzes the behavior of the exact sequence under the assumption that $X$ is a resolution of singularities of a normal surface.
• Uses the exact sequence to demonstrate that the localization sequence for ordinary Chow groups does not extend to Chow groups with modulus.
• Applies duality and duality theorems in $K$-theory to verify the exactness and structural properties of the sequence.

Experimental results

Research questions

• RQ1 Does there exist a natural exact sequence connecting $CH_0(X,D)$ and $K_0(X,D)$ for a smooth surface $X$ and effective Cartier divisor $D$?
• RQ2 Can the exact sequence be used to resolve the question of Kerz and Saito regarding $CH_0(X,D)$ when $X$ is a resolution of a normal surface?
• RQ3 To what extent does the localization sequence for ordinary Chow groups fail to extend to Chow groups with modulus?
• RQ4 How do the algebraic $K$-groups $K_0(X,D)$ reflect the geometric structure of $X$ and $D$ in the context of modulus cycles?
• RQ5 What structural differences exist between ordinary Chow groups and Chow groups with modulus on surfaces?

Key findings

• An exact sequence is constructed that connects $CH_0(X,D)$ and $K_0(X,D)$ for a smooth surface $X$ and effective Cartier divisor $D$.
• The exact sequence provides a positive answer to a question posed by Kerz and Saito when $X$ is a resolution of singularities of a normal surface.
• The paper demonstrates that the localization sequence for ordinary Chow groups does not extend to Chow groups with modulus, highlighting a fundamental structural difference.
• The exact sequence reveals a deep connection between $K$-theory and cycle theory in the context of modulus conditions.
• The failure of the localization sequence in the modulus setting is shown to be a direct consequence of the non-triviality of $K_0(X,D)$ in relation to $CH_0(X,D)$.
• The results confirm that Chow groups with modulus are not governed by the same formal properties as ordinary Chow groups, particularly in terms of localization.