In the lecture “Motives and ring stacks” Peter Scholze begins by saying that there are many ‘Weil-type’ cohomology theories for (separable and finite type) schemes over $\mathbb{Z}$ each of which can be realized as coherent cohomology of an associated ring stack defined over the coefficient ring $R$ of the cohomology theory. Examples he gives are Betti, de Rham and crytalline/prismatic. $\DeclareMathOperator{\Mot}{{Mot}}$
He then goes on to construct a presentable symmetrical monoidal $(\infty, 2)$-category $\Mot_{\mathbb{Z}}$ of motives for schemes over $\mathbb{Z}$. Category $\Mot_S$ of motives over any scheme $S$ can be construed as the category of modules over the motive of $S$ in the $(\infty, 2)$-category.
It is not yet clear to me what the relationship between the ring stacks “representing” the cohomology theories and these motives is. So my first question is:
- Given an appropriate cohomology theory $H(.,R)$ with coefficients in $R$, what is the precise relationship between $\Mot_\mathbb{Z}$ and the ring stack, $\mathcal{H}_{/R}$, associated to $H(.,R)$? Can $\mathcal{H}_{/R}$ somehow be built from $Mot_\mathbb{Z}$ and $R$ (as an analytic stack?)
Furthermore,
- Can there be a ‘universal’ ring stack $\mathcal{H}_{/\mathbb{Z}}$ over $\mathbb{Z}$ from which the ring stacks for various cohomology theories can be constructed?
Also,
- If $Sp_H$ is the spectrum representing cohomology theory $H$ in the stable motivic homotopy category, how is $Sp_H$ related to the ring stack $\mathcal{H}$ associated to $H$, and to $Mot_\mathbb{Z}$?
Finally,
- How does this category of motives stand in relation to Grothendieck’s conjectural abelian category of motives? Does it give a good definition of the Tannakian motivic Galois group?
