TL;DR
This paper extends Milne's theory of Lefschetz motives to more general settings, showing that algebraic and numerical equivalences coincide and providing a new approach to the associated Tannakian group.
Contribution
It generalizes the theory of Lefschetz motives to broader equivalence relations and base fields, and offers a novel, more conceptual computation of the Tannakian group.
Findings
Algebraic and numerical equivalences agree in this context.
Categories enjoy properties predicted by standard and less standard conjectures.
Provides a new, conceptual method to compute the Tannakian group.
Abstract
We develop Milne's theory of Lefschetz motives for general adequate equivalence relations and over a not necessarily algebraically closed base field. The corresponding categories turn out to enjoy all properties predicted by standard and less standard conjectures, in a stronger way: algebraic and numerical equivalences agree in this context. We also compute the Tannakian group associated to a Weil cohomology in a different and more conceptual way than Milne's case-by-case approach.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology · Advanced Algebra and Geometry