Construction of triangulated categories of motives using the localization property

TL;DR

This paper constructs a triangulated category of motives over quasi-projective schemes using the localization property and establishes the Grothendieck six operations formalism, also providing an integral étale realization of motives.

Contribution

It introduces a new construction of motives over quasi-projective schemes and proves the Grothendieck six operations formalism, including an integral étale realization.

Findings

Constructed a triangulated category of motives over quasi-projective T-schemes.

Proved the Grothendieck six operations formalism for these motives.

Developed an integral étale realization of motives.

Abstract

Using the localization property, we construct a triangulated category of motives over quasi-projective T-schemes for any coefficient where T is a noetherian separated scheme, and we prove the Grothendieck six operations formalism. We also construct integral \'etale realization of motives.