TL;DR
This paper introduces a category of motives for semi-algebraic spaces and demonstrates its triviality, revealing limitations in extending algebraic de Rham cohomology to these spaces.
Contribution
It defines a new category of motives for semi-algebraic spaces and proves its triviality, highlighting fundamental obstructions in the theory.
Findings
The category of motives for semi-algebraic spaces is trivial.
No suitable extension of algebraic de Rham cohomology exists for semi-algebraic spaces.
The period isomorphism cannot be maintained in this context.
Abstract
We define a category of motives for semi-algebraic spaces and show that it is trivial. This implies that there is no good extension of algebraic de Rham cohomology to semi-algebraic spaces compatible with the period isomorphism.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra · Algebraic Geometry and Number Theory