K-theoretic Chow groups of derived categories of schemes-on a question by Green-Griffiths

TL;DR

This paper introduces K-theoretic Chow groups for derived categories of schemes, extending classical concepts and identifying tangent spaces with cohomology, thereby generalizing known results to thickenings of schemes.

Contribution

It defines K-theoretic Chow groups for derived categories of schemes, proves their functoriality, and extends classical results like Bloch-Quillen to thickenings, answering Green-Griffiths' question.

Findings

K-theoretic Chow groups agree with classical ones for regular schemes

Tangent spaces are identified with cohomology groups of differentials

Extended Bloch-Quillen and Soule's variants to scheme thickenings

Abstract

Based on Balmer's tensor triangular Chow group, we propose K-theoretic Chow groups of derived categories of noetherian schemes and their Milnor variants for regular schemes and their thickenings. We discuss functoriality and show that our Chow groups agree with the classical ones for regular schemes. Moreover, we also define tangent spaces to our Chow groups as usually and identify them with cohomology groups of differentials. As an application, we extend Bloch-Quillen identification and soule's variant from regular schemes to their thickenings. This gives a positive answer to a question by Green-Griffiths.