Recollements for derived categories of enriched functors and triangulated categories of motives

TL;DR

This paper explores the categorical structure of Voevodsky's triangulated motives, establishing recollements for related enriched functor categories and their derived categories, and introduces the Voevodsky property for Serre localizations.

Contribution

It introduces the Voevodsky property for Serre localizing subcategories and applies it to embed Voevodsky's motives into recollements of derived categories.

Findings

Recollements for Grothendieck categories of enriched functors are established.

The Voevodsky property for Serre localizations is introduced and utilized.

Voevodsky's triangulated categories of motives are shown to fit into specific recollements.

Abstract

We investigate certain categorical aspects of Voevodsky's triangulated categories of motives. For this, various recollements for Grothendieck categories of enriched functors and their derived categories are established. In order to extend these recollements further with respect to Serre's localization, the concept of the (strict) Voevodsky property for Serre localizing subcategories is introduced. This concept is inspired by the celebrated Voevodsky theorem on homotopy invariant presheaves with transfers. As an application, it is shown that Voevodsky's triangulated categories of motives fit into recollements of derived categories of associated Grothendieck categories of Nisnevich sheaves with specific transfers.