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Mixed
AG (Algebraic Geometry)
Jannsen: LNM 1400 (Mixed motives and algebraic K-theory).
See reading notes from André, chapter 14.
Nekovar’s survey on the Beilinson conjectures (see section 3)
In Toën’s AIM talk on Homotopy types of algebraic varieties, there is a homotopical approach to the absolute/geometric dichotomy. File available here
There might be a vague intuitive idea saying that Deninger’s conjectural cohomology should be the geometric theory corresponding to the absolute theory which is Weil-etale cohomology. One could also ask if Arakelov motivic cohomology could be viewed as an absolute theory and if so, what is the corresponding geometric theory??
In nature, some cohomology theories in algebraic geometry seem to come in pairs, where one cohomology is referred to as “absolute” (or “arithmetic”) and the other is referred to as “geometric”. This is explained in section 3 of Nekovar’s survey article on the Beilinson conjectures (available here as well as in Motives vol I).
Examples of absolute cohomology theories should include Continuous etale cohomology, Absolute Hodge cohomology, Syntomic cohomology, and Motivic cohomology?
See also Arithmetic cohomology, Geometric cohomology
Remark: The term “absolute cohomology” was sometimes used back in the 80s for what we today call motivic cohomology.
nLab page on Absolute cohomology