Some properties that a cohomology can have: \begin{itemize} \item Respecting birational morphisms (between smooth projectives???)? \item Vanishing on affines? For Zariski topology, does sheaf cohomology with coeffs in a complex of sheaves also vanish on affines? \item Preserving limits and colimits? \end{itemize}
Rigidity, transfers, PD, versions of purity, Gersten conj.
Universality wrt some properties.
All suggestions. Universal theories (cf complex cobordism, and perhaps singular cohomology???)
Include Cisinski-Déglise on mixed Weil
Note that a projective bundle formula seems to allow us to compute cohomology of Grassmannians (see e.g. Levine and Krishna, end of section 5). This might be relevant when comparing two CTs, using Grassmannians as “generators” for a geometric cat.
Rognes email: Du spurte om “closed Mayer-Vietoris”. Det forekommer som Theorem 9.8 i Thomason-Trobaugh (Grothendieck Festschrift Volume III, side 379), som igjen bygger på artikler av Weibel.
Quote from Levine and Krishna section 6: “The additive Chow groups quite clearly do not satisfy the homotopy invariance property; for this reason it is difficult to apply the existing technology to prove either a localization property with respect to a closed immersion, or a Mayer-Vietoris property for a Zariski open cover. However, since we have a blow-up exact sequence, we can use the machinery of Guill´en and Navarro Aznar to define “additive Chow groups with log poles”, at least if we assume k admits resolution of singularities.”
Some theories have extra structure like Galois action or Hodge structure. When does a theory acquire a Galois action? For example: Does etale K-theory have one? Does the machinery of Quick relate to this at all?
Should understand the issue of torsion, and statement like (all CTs in Algebraic Topology are equivalent when tensored with Q“ and Morel’s statement on ”tensoring with Q makes the stable homotopy category give no info that’s not already there in the category of (Chow???) motives“.
Weil cohomology is tensored with Q, and motives is the universal such, roughly. Can we consider the universal theory with torsion??? This is probably complicated, but interesting.
nLab page on E10 Dictionary of properties of CTs in AG