Motives

TL;DR

This work develops a universal framework connecting universal cohomology theories to a conjectural theory of motives in algebraic geometry. It constructs universal and relative Künneth cohomologies $U_\kappa$ and $U_{\partial\kappa}$ via Levine's method, additive completion, and localisation, and shows how any cohomology $H$ in a Grothendieck-type class $\mathcal{S}$ realises from these via tensor functors, leading to notions of effective motives and their equivalences. In favorable settings the framework recovers known motive theories (e.g., Nori, André) and provides criteria for when a pure motive theory exists, tying the universal theory to the standard and nilpotence conjectures and yielding a Tannakian structure with a potential common fibre functor across Weil cohomologies. The standard hypothesis further implies that $Z_+$ acts as a universal coefficient field and forces coherence among Weil cohomologies, thereby unifying arithmetic and geometric invariants across characteristics.

Abstract

Making a survey of recent constructions of universal cohomologies we suggest a new framework for a theory of motives in algebraic geometry.

Theorems & Definitions (24)

• Theorem 1.1
• Proposition 1.2
• Example 1.3
• Proposition 1.4
• proof
• Example 1.5
• Theorem 2.1
• proof
• Proposition 2.2
• Example 2.3