Holmstrom 6 Memo notes Levine

Go through in detail when rewriting Mixed motives. Here are some stuff that should be looked at for other parts of the database:

References after the first conjecture, to L-functions and geometric applications.

Ref to K-theory including Bloch’s famous lectures.

Mention of absolute cohomology, and Lichtenbaum’s conjectures.

Def and fundamental properties of higher Chow groups.

Suslin homology and Friedlander-Suslin cohomology

Also cubical Chow groups and comparisons.

Chapter 3:

Def of Bloch-Ogus cohomology.

Mixed Tate motives.

Motives through realizations, Jannsen, Deligne, perhaps Huber (or is this only triangulated?), …

Stuff on cdga’s.

Section 6: Cycle classes, regulators, realizations.

Motivic cohomology as the universal Bloch-Ogus theory. Using relative Chow groups.

Regulators: Goncharov described regulator to real Deligne cohomology. Kerr refined this to integral Deligne cohomology in “A regulator formula for Milnor K-groups” and “The Abel-Jacobi map for higher Chow groups” with Lewis and Muller-Stach.

Get cycle class map from K-theory to any Bloch-Ogus theory, factoring through motivic cohomology.

Want realization functor on $DM(k)$ for any Bloch-Ogus theory. To do this, we should consider the following notion:

Definition of “geometric cohomology theory”, a notion refining the Bloch-Ogus axioms. Examples: de Rham cohomology, singular cohomology, étale cohomology with mod n coeffs. Nonexamples: absolute Hodge cohomology, Deligne cohomology, $\ell$-adic cohomology.

For the nonexamples, can still construct realizations, see Levine: Mixed motives.

Also several other approaches and problems related to realization functors on various cats of motives, for various CTs.

Rererences (go through!)

nLab page on 6 Memo notes Levine