Dimensional homotopy t-structure in motivic homotopy theory

TL;DR

This paper constructs and analyzes homotopy t-structures in motivic homotopy theory, extending foundational work and providing new tools for understanding motives and their homology.

Contribution

It introduces new homotopy t-structures with desirable properties, extending the theory of homotopy invariant sheaves and relating to Gersten weight structures.

Findings

Computed homology of motives, especially for relative curves

Established properties of the new t-structures analogous to perverse t-structures

Extended the theory of homotopy invariant sheaves with transfers

Abstract

The aim of this work is to construct certain homotopy t-structures on various categories of motivic homotopy theory, extending works of Voevodsky, Morel, D\'eglise and Ayoub. We prove these $t$-structures possess many good properties, some analogous to that of the perverse $t$-structure of Beilinson, Bernstein and Deligne. We compute the homology of certain motives, notably in the case of relative curves. We also show that their hearts provide convenient extensions of the theory of homotopy invariant sheaves with transfers, extending some of the main results of Voevodsky. These t-structures are closely related to Gersten weight structures as defined by Bondarko.