Holmstrom Old notes I

Scholl says that there are many examples of smooth schemes, both higher-dimensional over Z, and curves over other rings of integers.

Levine conversation:

• Talk to Alexis Bouthier, about his Mast thesis and de Rham cohomology, maybe also about transfers.
• The main potential problem is that Deligne cohomology corresponds to a nonconnective spectrum
• Ivorra thesis: Godement resolution vs correspondences
• To pass form unstable to stable, you need something which I think he called P1-delooping. This you get form localization/Gysin sequence.
• Nisnevich descent should follow from Gysin seq. No need to use transfers here unless you want to work in DM rather than SH.
• A technical problem for Deligne cohom representability: Extend finite corr to compactifications, maybe prove independence of choices or something.

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Outline of idea, presented to Scholl after Oberwolfach:

Gillet suggests somewhere that arithmetic Chow groups should be expressed as sheaf cohomology with coeffs in a sheaf of DGAs. Since DGAs are a baby version of A-infty algebras, it is natural to look for a generalized sheaf cohomology with coeffs in a sheaf of E-infty algebra, or more generally a sheaf of (E-infty) ring spectra. Intuition here is to think of sheaf cohom as Hom in homotopy cat. Waht homotopical setting to use??? Motivic stable homotopy theory should be a natural first try. Get ring spectrum here from pushforward from SH(Spec(Q)). All the four functors in Ayoub's formalism are supposed to respect cofiber sequences, and map ring spectra to ring spectra.

To produce the ring spectrum over Q, modify Feliu's construction a little. She works with zigzag of Zariski simplicial sheaves, I think.

Things to check:

• A1-invariance
• Nisnevich descent?
• No problem because of zigzag
• Check stable representability
• Check that the products on cohomology groups "lift" to an E-infty structure on the spectrum
• Check potential problem with nonconnectivity of Deligne spectrum
• Check pushforwards, maybe via Panin et al formalism
• Check for which schemes this gives the right thing

Panin ref: K-th archive 619. Maybe also "On Voevodsky's algebraic K-th spectrum" with Roendigs and Pimenov.

For the class of schemes over Z, it might be relevant that the object of SH corresponding to a smooth scheme is compact (ref Ayoub).

Idea: Maybe we could do something special with the bad fibers, i.e. use the above construction but just modify it to take care of these fibers. Maybe use something like the h-topology.

Probably irrelevant: For existence of f lower shriek, Morel (Basic prop p 10) requires f to be smooth, and Ayoub requires quasi-projective.

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Roendigs conversation 28 July 2009

I asked about the existence (and uniqueness) of a ring spectrum E over Z given prescribed pullbacks to Q and to Fp for all finite primes p.

See the photo of the blackboard for some triangles he wrote down, that did not seem directly related to the actual questions.

First main point: Motivic cohomology might possibly not behave wrt pullback in the sense that there is a candidate for the motivic cohomology spectrum over Z, but it is not known that it pulls back to what one generally believe is the correct thing over Fp. K-theory is better behaved in this respect, so maybe arithmetic K-theory is better to look at then motivic cohom. 2nd main point: Panin style formalism for pushforwards should work over a general regular base scheme. No reference given, maybe there exists one.

Ask Ayoub, and also Deglise-Cisinski, about the initial question.

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Understand Jakob’s spreading out functor (“intermediate extension”). Could it be an alternative to glueing? Namely, just spread out the spectrum over Q to one over Z??

nLab page on Old notes I