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I am reading Mazza's, Voevodsky's and Weibel's book Lecture Notes on Motivic Cohomology and have grown curious about the following question:

Which cohomology theories on $Sm/k$ are representable, i.e. appear as Hom-groups, in Voevodsky's construction of $DM_{\text{Nis}}^{eff,-}(k, R)$?

If we let $X$ be a scheme of finite type over $k$ and $R$ a commutative unital ring, then we have (by construction) motivic homology with coefficients in $R$ $$H_{n,i}(X, R) = \text{Hom}_{DM_{\text{Nis}}^{eff,-}}(R(i)[n], R_{\text{tr}}(X)),$$ and motivic cohomology $$H^{n,i}(X, R) = \text{Hom}_{DM_{\text{Nis}}^{eff,-}}(R_{\text{tr}}(X), R(i)[n]).$$ From this, we obtain the algebraic singular homology (and hence Suslin's singular homology) as $$H^{\text{sing}}_n(X,R) \cong H_{n,0}(X,R)$$ and, if $X$ is a smooth separated scheme of finite type over some perfect field $k$, the higher Chow groups $$CH^q(X, 2q-p) = H^{p,q}(X,\mathbb{Z}).$$

It is clear to me that such cohomology theories must satisfy certain properties:

  • Mayer-Vietories
  • $\mathbb{A}^1$-invarience
  • Künneth-theorem
  • ...

But do we have any other examples of cohomology theories that appear as Hom-groups in $DM_{\text{Nis}}^{eff,-}(k, R)$? How about

  • Betti cohomology?
  • $l$-adic cohomology?
  • Crystalline cohomology?
  • Algebraic de Rham cohomology?

Another way of framing the question is: How far off is Voevodsky's-Morel's $DM_{\text{Nis}}^{eff,-}(k, R)$ construction from being the category of pure/mixed motives in the sense of Grothendieck.

(I realize this might be a very hard question to which not much is known.)

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    $\begingroup$ Another "obvious" requirement is that the cohomology theory must have Voevodsky transfers (i.e. factor through the category $\operatorname{Corr}_k$ of Voevodsky correspondences). $\endgroup$ Feb 27 '21 at 16:10
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Recall that $\mathrm{DM}(k)$ can be described as the subcategory of $\mathrm{SH}(k)$ made of modules over the motivic cohomology. This implies that cohomologies which are representable in $\mathrm{DM}(k)$ receive a cycle class map, so they must admit additive Chern classes ($c_1(L\otimes L')=c_1(L)+c_1(L')$). In fact, I think this is an if and only if, in the sense that motivic cohomology is the universal among cohomologies with additive Chern classes, but I do not know right now if this is written anywhere in the motivic setting with integral coefficients. In any case, if you put $R=\mathbb{Q}$ then you can have any type of Chern classes, for example $K$-theory. This is all written at Cisinki-Déglise's Triangulated categories..., chapter 14, which is in general a great reference for many motivic results up until 2009.

Your question is more natural for $\mathrm{SH}(k)$, as it happens already in topology, and then you know your cohomology would belong to $\mathrm{DM}(k)$ depending on if it is a module over motivic cohomology or not. An answer to your question in the setting of $\mathrm{SH}$ is in the paper of Cisinski and Déglise, Mixed Weil cohomologies, at 2.1.5. Loosely speaking, for any "sheaf cohomology" they construct an object of $\mathrm{SH}(k)$ representing such cohomology. Their hypothesis must imply having additive Chern classes, since they are willing to define a mixed variant of Weil cohomologies (which have additive Chern classes), so their cohomologies are in $\mathrm{DM}(k)$. At the end of the paper you have their main examples.

I recall that Cisinski-Déglise's method was later rewritten and developped by Holmsotrom-Scholbach in Arakelov motivic cohomology, section 3, to represent the Deligne cohomology, and by Déglise-Mazzari in The rigid syntomic spectrum at 1.4.10 where they proved that the rygid syntomic cohomology is representable.

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    $\begingroup$ $DM$ is absolutely not a full subcategory of $SH$. It is, as you correctly mention, a category of modules. $\endgroup$ Feb 27 '21 at 10:37
  • $\begingroup$ Shame on me! ^_^' Thanks a lot for pointing out my mistake $\endgroup$
    – Tintin
    Feb 27 '21 at 11:21

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