Deglise and Cisinski has a finiteness theorem on p 8 of Triangulated preprint, related to mixed Weil cohomologies.
Bondal and Van den Bergh: Generators etc. Some results on finiteness of global Exts, I think. This paper is mentioned in Toen: Champs affines.
arXiv:1103.5544 Finite generation conjectures for cohomology over finite fields from arXiv Front: math.KT by Thomas H Geisser We construct an intermediate cohmology between motivic cohomology and Weil-etale cohomology. Using this, the Bass conjecture on finite generation of motivic cohomology, and the Beilinson-Tate on the finite generation of Weil-etale cohomology are related.
Dan Petersen MO answer on realizing homotopy types by finite topological spaces
arXiv:1205.6148 DG categories and exceptional collections from arXiv Front: math.AG by Agnieszka Bodzenta Bondal and Kapranov describe how to assign to a full exceptional collection on a variety X a DG category C such that the bounded derived category of coherent sheaves on X is equivalent to the bounded derived category of C. In this paper we show that the category C has finite dimensional spaces of morphisms. We describe how it behaves under mutations and present an algorithm allowing to calculate it for full exceptional collections with vanishing Ext^k groups for k > 1. Finally, we use it to describe an example of a non-commutative deformation of certain rational surfaces.
Conj: Br X finite for X proper over Spec Z. This would imply finiteness of Sha.
Maybe one can use Szymik’s Brauer spectra? This provides a link between the group of units, Pic, and Br. The construction could maybe be extended to non-affine schemes via Thomason’s hypercohomology construction.
Hope: use interaction between various types of homotopy types? What about the case of schemes over Fp - is this easier? Can we use Noetherian induction, or induction on dimension? See Kahn for reformulations of Bass conjecture, some version of this could maybe also apply for Brauer groups and motivic cohomology. Note difference between etale motivic cohomology and motivic - is the etale version easier to handle? Use also Serre classes. Consider classes of spectra defined by finiteness conditions on homotopy groups, one could study the inverse image under an infinite loop space machine of such classes. Can we prove that the class of symmetric monoidal categories whose spectra have finitely generated homotopy groups are closed under some constructions???
For reducing general questions about K-theory of schemes to the case of fields, one often uses the localization theorem of Quillen or Thomason-Trobaugh. Can we apply this either to finiteness properties, or to Arakelov motivic cohomology?
There is a Picard scheme - is this relevant for finiteness of Pic? Is there also something like a Brauer scheme? What about other cohomology invariants - when are they associated to a moduli scheme?
Somewhere there is some kind of adelic moduli interpretation of Pic. Again, can this be generalized??? Check the Ribet volume maybe???
Is there any chance of using ideas from rational homotopy theory? For example, if a manifold is elliptic (in the sense of the rational homotopy theory), then I think both the rational cohomology and rational homotopy are both finite dimensional). See also my MO question on finiteness of stable homotopy groups of spheres.
Could one use Thomason’s descent spectral sequence to relate the naive presheaf cohomology groups of Gm to the etale (or Nisnevich) cohomology groups of Gm? The first should be finite groups. Does it help to have the model structures of Barnea-Schlank?
For example: Could it be that when we look at this spectral sequence, things seem to not stay within the realm of finite groups (or finitely generated groups) but that we can prove the existence of a weak fibration structure on some class of (pre)sheaves of spectra WITH SOME FINITENESS CONDITION so that, as a consequence, there exists a (globally???) fibrant replacement of Gm satisfying the same finiteness condition. That would be awesome. I am quite sure finitely generated is true for the naive presheaf groups of Gm applied to an arithmetic scheme.
Try to use devissage (as mentioned in Kahn’s article in relation to B-S vanishing) to prove vanishing of H-hat groups.
If working on Weil-etale stuff, finiteness problems would probably be a great topic. For example, assuming a six functors framework, can I prove finiteness theorems, and in particular finiteness of Sha? Study first the relation between six functors in l-adic setting and finiteness theorems for etale cohomology.
Friedlander chapter 7 has a statement about the etale homotopy groups of essentially any (rigid) hypercovering begin finite groups.
Blog post by Goss mentioning the use of coherent cohomology (maybe in the setting of rigid geometry) to show that spaces of modular forms are finite-dimensional.
http://mathoverflow.net/questions/33042/countable-hom-ext-implies-finitely-generated
Question: Can we use homotopy theory to prove finiteness results in arithmetic geometry? (Morel)
Déglise and Cisinski has a finiteness theorem on p 8 of Triangulated preprint, related to mixed Weil cohomologies.
Bondal and Van den Bergh: Generators etc. Some results on finiteness of global Exts, I think. This paper is mentioned in Toen: Champs affines.
arXiv:1103.5544 Finite generation conjectures for cohomology over finite fields from arXiv Front: math.KT by Thomas H Geisser We construct an intermediate cohmology between motivic cohomology and Weil-etale cohomology. Using this, the Bass conjecture on finite generation of motivic cohomology, and the Beilinson-Tate on the finite generation of Weil-etale cohomology are related.
nLab page on Finiteness theorems