Holmstrom Representable CTs

See file Representable.tex

Old suggestion: Do the L-construction in the second paper, and the easier construction in the first. Or use the L-construction in the BO paper to get spectra where there are no products at all on the complexes.

There was a pre-Oberwolfach misunderstanding somewhere, coming from believing that the L-construction was necessary to get a ring structure, while in fact it seems like the only advantage of the L-construction is to be slightly more canonical, and possibly give a spectrum in the setting where we don’t have a product.

Explain the problem of passing from a strict ring structure on the complex computing hyper cohomology to a ring structure on a resolution. Solutions: Thom-Sullivan (Levine, Huber?) or infinity-cat for integral coeffs, DC model structure for rational coeffs.

This model structure (for which one would have to check tractability) turns out not to be needed, since the Burgos complexes are fine sheaves. If we needed to take this kind of resolution (G-replacement), one would have to view all these complexes as complexes on the big site, take G-replacement, and then use a statement saying that cohomology on the small and the big site agree, see for example Milne, III.3 OSLT.

WARNING: There might be a problem here with functoriality, related to the compactifications of smooth varieties. This has to be worked out. Jakob: this is ok. I am still confused by a related thing though: the fact that we take hypercohomology on the compactification. How can this give us sheaves on the big site of smooth things, even if we pass to a limit?

Cisinski Tuesday discussion: For checking that something is an Omega-spectrum, one needs three things: A1-invariance, etale descent (weird, I thought this was Nis descent), and weak Kunneth formula. The last thing means the following:

(1)

\forall i, q, X: H^i(X \times \Pone, \cE(q) ) \cong H^i(X, \cE(q) ) \oplus H^{i-2}(X, \cE(q-1) )

Note that if you know projective bundle formula for a point, then weak Kunneth is equivalent to the general projective bundle formula. Proof of weak Kunneth: Prove it directly, or via the existence of Chern classes (or maybe a map from Pic), see the Gysin paper for details.

We have $c \in H^2( \Pone, 1) = Hom_{\DMAeff}(M(\Pone), E^{(1)}[2])$ , where $M(\Pone) \simeq \mathbf{K} \oplus \mathbf{K}(1)[2]$ , and in fact $c$ corresponds to a map from $\mathbf{K}(1)[2]$ as can be seen from the fact that it is zero when restricted to any point of $\Pone$ , or from the formula

(2)

H^{*}(\Pone, q) = H^*(\Spec(k), q) \oplus H^{ * - 2}(\Spec(k), q-1) \cup c

See also p D18 for more details.

About etale descent: There are two different definitions of descent, right? And when using the simpler definition (in terms of Cech coverings only I think), did we then exploit the fact that we were dealing with etale descent rather than some other form of descent? (In the notes there is a diagram which appears to start with a general hypercovering OSLT, and then replace it with a Cech type hypercovering, OSLT, with an accompanying note that the augmentation map is etale.

I think Fulea gives an associative and commutative product on the level of the Deligne complex of sheaves (i.e. we must take hypercohomology of this to get Deligne cohomology). Levine also said he knows how to do this, by “averaging with certain binomial coefficients”. To get functorial complexes like we need we must take a resolution of our complex and then global sections. There is a problem here in proving that the resolution also has a strictly associative and commutative ring structure. There should be at least two ways to resolve this problem. One is by having a model structure on sheaves of CDGAs such that weak equivalences are defined by underlying weak equivalences on complexes, and FIBRATIONS forget to FIBRATIONS (right Quillen?) (or something like this; this will assure us that cofibrant replacement inside CDGAs is weakly equivalent to cofibrant replacement (i.e. resolution) inside complexes. The other solution is the following: Take the Godement resolution, it has a cosimplicial structure, and there is a general method (Thom-Sullivan construction) for getting a string ring structure on any such gadget, see Levine’s big book on mixed motives from page 275.

There is something still to understand related to the possibility of having negative twists: Do you then have a different definition of spectrum, with a “negative” symmetric group action? I really don’t think so: I think nonzero cohomology groups with negative twists can occur even with the standard definition of spectrum. Or what do you do? Compare with topology where you can have nonconnective spectra, see maybe Lurie’s survey for definition of spectrum, or some other topology source.

\subsection{Other remarks related to the Deligne spectrum}

I would like to understand the point mentioned by Cisinski and Deglise in their email, saying that every Bloch-Ogus cohomology is representable by a spectrum. Deglise referred to Levine chapter 5.

It might be interesting to say something about absolute Hodge cohomology. I believe the absolute Hodge spectrum can be constructed as the connective cover of the Deligne spectrum. Cisinski said that connective cover makes sense, and gave a reference to some construction of Voevodsky (the homotopy t-structure). Whether connective means nonpositive or nonnegative vanishing depends on whether you use homological or cohomological convention for motives.

Could it happen that with two different complexes both computing Deligne cohomology, one satisfies Nisnevich descent and the other one doesn’t? NO, see holim def or descent, and holim is compatible with quasi-isos.

We might also need “stability” in the sense of DC, but a modified notion which takes into account that Tate twist might not be an isomorphism. Cisinski said that Deligne cohomology is stable, and that stability is a consequence of the projective bundle formula. It would be nice if we could explain this in detail, and maybe make a more general statement, and then get Deligne cohomology as a special case. Which theories satisfy stability in general - give examples of some that do and some that don’t. Is it all ordinary theories? What about if we work with integral coefficients - can you make sense of stability then? Statement of page 121: Adjunction between SH and $D_{\mathbb{A}^1}$ , where the latter should be thought of as the universal derived premotivic category satisfying homotopy, stability and Nis descent. Could one make a statement for general Bloch-Ogus theories, with some additional hypotheses maybe? Something to understand here is why we get different types of spectra (i.e. $\mathbb{P}^1$ vs $\mathbf{K}(1)$ ) in different situations). Can one pass freely between different types of spectra, by Quillen equivalences maybe, and between monoids in the different categories of spectra?

This is probably not needed, but could be good to remember for the future: I was first thinking that the following sequence would give the Deligne complex: I though Fulea gave a complex of sheaves, which we could think of as being on the big site, then taking G-fibrant replacement, then global sections, then F-invariants. This should compute Deligne cohomology. Here we would have used that cohomology on the small and big site agree (Milne III.3 I think).

Should explain here or later in what way precisely the L-approach is more canonical. Maybe another advantage is that one can construct spectra for CTs without products. Cisinski discussion: Using the canonical bundle gives a cohomology class c etc, this is canonical at least up to a sign. The process here is equivalent (?) to

(3)

\R = H^0(pt, 0) = Hom(K, E) = Hom(\Kone[2] , E^{(1)}[2])

The L-approach is needed to remove choice of lift to the model category, not really for the choice of the cohomology class c.

Note to us: At some point we might need that sheafification of a monoid is a monoid: this is because sheafification is exact, and tensoring with an object is computed, at least on representables, by something related to categories and products in Sets. Ask Peter if there is a problem with understanding this.

Note that if two different complexes are sectionwise quasi-isomorphic, then one is fibrant iff the other is. This follows from the definition of t-descent in terms of hypercoverings, and that holim is compatible with quasi-isomorphisms (make this precise).

Orientability of Deligne cohomology: The easiest way to prove this is probably via etale descent. Deglise said that with rational coefficients, Nisnevich and etale descent are equivalent. One way of making this precise is the equivalence $DM_{et}(k) \otimes \Q \simeq DM(k) \otimes \Q$ . Cisinski also said (I think) that it would be immediate from the existence of Chern classes.

Notes from Huber discussion: Maybe an easier way to get a ring structure could be to take holim in a good model structure (i.e. the G-stable one for G the representables) of the diagram

(4)

\Z \to D \gets \Omega^*

maybe in some category of complexes of sheaves or in some sort of Hodge complexes. Not completely sure how this gives a product structure on D though. Maybe I misunderstood? Can the diagram be used in some other way? Hmm, probably the diagram should have $\Omega^*$ in the middle and the $F^p$ part of the Hodge filtration on the right.

We might hope to get to integral coefficients and/or weak Hodge by using some form of Hodge complexes, maybe following Huber (see handwritten note).

Note: The idea of an integral lattice in Deligne cohomology is related to the integral structure on de Rham cohomology, which only makes sense over Spec Z, not over Q. What about over a different number ring?

Cisinski said that it makes sense to talk about connective motivic ring spectra (Non-positive or non-negative vanishing depends on whether you use homological or cohomological convention), and the connective cover of a strict ring spectrum (which should be the right adjoint to the inclusion of connective ring spectra into all ring spectra). The tool for expressing this is the homotopy t-structure.

At some later stage, maybe in a paper on higher arithmetic RR, we could spell out in more details the general consequences of Ayoub’s formalism, functoriality, dualities, etc, why you need purity and and orientability and so on, what works over a field and what works over a one-dimensional base (I think purity is open in general in the latter case, but known maybe in some cases, and for 2-dimensional bases I think there is a counter-example by Ayoub or Morel, but maybe I am confusing this with something else).

Small insight: Maybe our first approach works only assuming products, but the L-approach works for non-ring spectra as well. Can we show that both give equivalent spectra in some sense? Must show in particular that the chosen map is nonzero. See further notes in the “future” pile.

Question: Is it true that any BO theory satisfies the projective bundle formula and/or weak Kunneth?

Orientability and Chern classes should correspond to proj bundle f. Should also get descent properties for all BO theories from this. What kind of descent can you prove with integral coeffs in general?

nLab page on Representable CTs

Created on June 9, 2014 at 21:16:13 by Andreas Holmström