One could hope to prove for example finiteness or vanishing theorems via some form of “descent” or “devissage” arguments, possibly involving various spectral sequences.
How about proving Beilinson-Scholbach duality?
In a letter of Grothendieck he discusses devissage arguments for finiteness, see p 299 of Motives vol 1. Here are some terms like “motif serein” that I don’t understand.
A CT in Algebraic Topology is determined by a (graded?) group. What is the exact relation between the category of CTs (the stable homotopy category) and the category of graded (periodic??? see Lurie) groups? Does phantom maps have anything to do with this, or is that something else? It seems “easy” to describe the category of graded (abelian) groups, so there is probably not an equivalence of cats. Does it matter that the category of filtered groups is not abelian? In principle, if we know the cohomology of a point, we know the cohomology of any space. Does this mean that we can prove for example that if the cohomology of a point if finitely gen, the same is true for the cohomology of any “finite space”, for example a finite CW cplx?
Is it true that CTs in Algebraic Geometry are determined by the cohomology of a point? Would this mean the cohomology of the base scheme or what? What can one say on the level of categories? For example, what is the relation between the category of CTs and the category of bigraded groups? Should we look at the category of bigraded groups with an action of some group (motivic Galois group?) Could we find an equivalence between the category of bigraded CTs and the category of bigraded groups with a (motivic Galois?) action, given by taking the cohomology of a point? What could be inferred about for example finite generation, from the knowledge of the cohomology of a point? What is a “finite space” in Algebraic Geometry? When is motivic cohomology expected to be finite?
In Algebraic Topology, a CT is determined by its value at a point. Is there a similar principle in Algebraic Geometry? For example, could we prove Hodge conjecture by showing that and a suitable algebraic cycle group agree on a point? Probably not, but perhaps there is some other class of geometric objects whose values determines the CT. Note that the rational motivic cohomology of a number field should be related to (determined by?) functional equations satisfied by real analytic versions of polylogarithms (see Goncharov 1995, review).
nLab page on Descent type ideas