Why is there only one grading in Algebraic Topology? Could it be that the twists are there but, like for étale cohomology, they do not affect the cohomology group? For “geometric CTs in Algebraic Geometry”, whatever this means, it seams like the twist carries only “arithmetic information”, so omitting it still gives a sensible geometric CT. Is there “arithmetic information” omitted in CTs in Algebraic Geometry, or is there simply no such information to consider? Note: a remark in one of the late 80s papers of Beilinson et al, this says this is related to algebraic K-theory being Z-graded and topological K-theory being Z/2-graded.
Some ideas are in the concept note.
This should include an understanding of what “homotopy type” means. A strange example: The homotopy type should determine all cohomology, in particular the motive, of for example an elliptic curve. However, the motive determines an L-function. Assuming that Severitt’s work apply to Q (which it may not), the homotopy type seem quite determined by the genus, but not all elliptic curves have the same L-function!
There is something curious: any (nice) scheme produces a suspension spectrum, and hence a CT. Any CT produces a sheaf. In some sense, schemes, spectra, CTs, motives, sheaves, are parts of the same thing, and to some extent interchangeable and combinable. Drinfled’s student said in a talk: A scheme is a kind of sheaf of sets.
On page 10 of [Algebraic Geometry over MCs], Toen suggests that some CTs can be defined in much generality for general geometric stacks, i.e. for algebro-geometric objects over for example symm monoidal infty-cats. E.g. étale K-theory of ring spectra.
In Arakelov theory, one wants to describe invariants of arithmetic surfaces, especially canonical classes and inequalities between these. See intro to de Jong: Explicit Arakelov geometry; such inequalities would imply effective Mordell, abc and Szpiro’s conjecture! Can these invariants be expressed in terms of cohomology/stable homotopy theory over ?
Follow up on Quick’s article? Compute more things? Compute things with Morel-Hopkins spectral sequence?
Quoting Levine in K-th handbook p 453: (talking about mixed motives and Bloch-Ogus theories) “It is not at all clear what relation K-theory has to the cohomology arising from these constructions”.
nLab page on Some ideas