Remarks on 1-motivic sheaves

TL;DR

This paper extends the construction of categories of 1-motives with torsion and 1-motivic sheaves to perfect fields, and shows their derived categories are equivalent over transcendental fields.

Contribution

It generalizes existing constructions of 1-motives and sheaves to a broader class of fields and establishes an equivalence of their derived categories in new cases.

Findings

Categories of 1-motives with torsion and 1-motivic sheaves are extended to perfect fields.

Proves the equivalence of their bounded derived categories over transcendental fields.

Enhances understanding of 1-motivic structures in algebraic geometry.

Abstract

We generalize the construction of the category of 1-motives with torsion ${}^tM_1$ (introduced by Barbieri-Viale, Rosenschon and Saito) as well as the construction of the category of 1-motivic sheaves ${\rm Shv}_1$ (defined by Barbieri-Viale and Kahn) to perfect fields $k$ (without inverting the exponential characteristic). For $k$ transcendental over the prime field we extend a result of Barbieri-Viale and Kahn, showing that ${}^tM$ and ${\rm Shv}_1$ have equivalent bounded derived categories.