See file aspaces.tex.
Goal: Try to understand the possible role of homotopy theory in understanding compactifications of arithmetic schemes, the field with one element, the ideas of Deninger and Flach-Morin, and conjectures on L-functions and zeta functions.
GOAL: Find the right notion(s) of a category of arithmetic spaces. This would be the “Arakelov” or “F1” cat corresponding to cats of simplicial sheaves over fields of char zero.
Gillet: Filtrations on higher alg Kth
See the dream of Deninger in Motives vol
Would like the cat to include compactifications of arithmetic schemes. Take into account the paper by Burgos et al describing stuff at the infinite place
Note: Can extend DM to simplicial schemes (Deglise in Oberwolfach talk). What about the six operations? This should be of interest in a complete reworking of SH and DM for generalized schemes, including the compactification of Spec(Z).
Note: Interesting material in Morin: Thesis chapter 1, plus Utilisation article in Compositio (2008?). These cover the arithmetic-topological dictionary and include comments on Deninger and Weil-etale.
Notes towards Toulouse talk April 2011:
What should the ultimate understanding look like, when it comes to cohomology of arithmetic schemes and their compactifications? Some key players:
H-hat could maybe be interpreted as cohomology with compact support with respect to the includion of Spec Z into its compactification. Deninger’s cohomology might be the geometric counterpart to the absolute theory H-hat. Maybe we can relate this picture to that over a finite field, to some extent, at least the absolute/geometric part, and the fact that special values seem to relate to the absolute cohomology, while the definition and more elaborate analytic properties apparently relate to the geometric cohomology.
One might also hope for the functional equation relating to some form of Poincare duality. Is PD in classical setting valid for geometric as well as absolute cohomology? What does the abstract PD in terms of six functors say, does it give a PD statement for any CT, like for example K-theory and G-theory?
Scholbach thesis: Analogy with PD on an affine curve, and with global duality in arithmetic.
Note that Weil-etale cohomology already exists for finite fields and behaves very well.
Mention cycle class map, and the possibility of this coming from a Hochschild-Serre spectral sequence of some kind. The cycle class map should be from H-hat to Weil-etale (according to Morin) but also to Deninger (according to Gillet-Soule in Motives), the latter being injective according to conversation with Soule.
Guess: Maybe Deninger’s cohom takes values in some ind/pro-Tannakian category whose derived category is related to or equal to the homotopy category of H-hat-modules. Could the Lurie theory of higher Tannakian stuff be the right thing to consider instead?
Sources to check: Scholbach’s thesis and paper. Deninger’s papers. The F1-literature. Morin/Flach/Geisser/Lichtenbaum maybe. Also Flach conversation notes.
Q: If the Galois action on geometric etale cohomology comes from the lower/upper star term in the Hom group expressing cohomology, could one hope that the same is true for Deninger’s phi action?
Scholbach: Pole orders should be given by negative Euler char of motivic cohom of the Verdier dual.
Note that Scholbach could not make the completed L-function triangulated (but noncompleted its ok, see thesis remark 5.2.2 and 5.5.2. The factor at infinity is not triangulated on the category of mixed Hodge structures, see FPR94, 1.1.9 and 1.2.5.
For Scholbach’s speculation, see thesis remark 6.1.2, page 116 in web version.
Notes from Gillet-Soule: Arithmetics analogs of the standard conjectures (note a page is missing in my file). Notion of arithmetic correspondence (in terms of regular kernels), which is too restrictive, this is related to the problem that there is no field with one element. Conjecture on the arithmetic intersection pairing being nondegenerate, some results apparently in the case of arithmetic surfaces. The full arithmetic Chow groups are analogous to usual Chow groups for smooth projective var over finite field. The cycle class remark is on page 135, just before section 3. It says that the FULL arithmetic Chow groups (tensored with the reals) should inject into Deninger’s infinite-dimensional cohomology. The latter should be analogous to l-adic cohomology.
Notes from Jannsen in Motives: PD in DM should take the form where R is the motive of a variety (all this is over a field, see the section 4).
Notes from Lichtenbaum in Motives vol: Poses questions on the existence of CTs describing the zeta function, its special values, values up to rational number, or only the zero/pole orders. Claims that for describing the zeta function itself, there is not even a conjectural answer. Furthermore, even over finite fields no answer giving the precise special values. Some axiomatics are given for complexes and , and the etale cohomology of these are conjectured to describe (in some sense) the zeta values for smooth projective varieties over a finite field. Interesting remark: should be the Brauer group of if is of dimension 2, and “the Zariski site holds no hope of accommodating the Brauer group”.
Deninger: Motivic L-functions and regularized determinants. Too much too digest in one skim-reading, must study more carefully with pen and paper.
nLab page on Arithmetic spaces