On extending Soul\'e's variant of Bloch-Quillen identification

TL;DR

This paper introduces Milnor K-theoretic Chow groups for derived categories, extending classical groups to detect nilpotent elements and applying this to generalize Soulé's Bloch-Quillen identification to infinitesimal thickenings.

Contribution

It defines new K-theoretic Chow groups for derived categories and extends Bloch-Quillen identification to infinitesimal thickenings, addressing a question by Green-Griffiths.

Findings

Milnor K-theoretic Chow groups recover classical groups for smooth projective varieties.

These groups can detect nilpotent elements that classical groups cannot.

Extended Soulé's variant of Bloch-Quillen identification to trivial infinitesimal thickenings.

Abstract

Based on Balmer's tensor triangular Chow group [2], we propose (Milnor)K-theoretic Chow groups of derived categories of schemes. These Milnor K-theoretic Chow groups recover the classical ones [6] for smooth projective varieties and can detect nilpotent, while the classical ones can't do. As an application, we extend Soul\'e's variant of Bloch-Quilled identification from smooth projective varieties to their trivial infinitesimal thickenings. This answers affirmatively a question by Green-Griffiths for trivial deformations, see Question 1.1 below.