Adjoint functors on the derived category of motives

TL;DR

This paper investigates the existence of adjoint functors in Voevodsky's derived category of motives, revealing which subcategory inclusions admit such functors and their implications for approximating motives.

Contribution

It provides a comprehensive analysis of the existence of adjoint functors for subcategories of motives, including new cases and obstructions related to Chow group finite generation.

Findings

Some adjoint functors exist, including previously unexplored ones.

Certain adjoint functors do not exist due to Chow group finite generation failures.

Exact conditions for existence of adjoints are determined for specific base fields.

Abstract

Voevodsky's derived category of motives is the main arena today for the study of algebraic cycles and motivic cohomology. In this paper we study whether the inclusions of three important subcategories of motives have a left or right adjoint. These adjoint functors are useful constructions when they exist, describing the best approximation to an arbitrary motive by a motive in a given subcategory. We find a fairly complete picture: some adjoint functors exist, including a few which were previously unexplored, while others do not exist because of the failure of finite generation for Chow groups in various situations. For some base fields, we determine exactly which adjoint functors exist.