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Let $k$ be a field. I would like a reference for realization functors from Morel-Voevodksy's stable category $SH(k)$ to the derived categories of $Gal(\bar{k}/k)$-modules. Has something like this been written down?

Thanks!

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The most general functor of this form was constructed by Ayoub in La réalisation étale et les opérations de Grothendieck.

Ayoub considers the ∞-category $DA^{et}(S,\Lambda)$ which is defined exactly as $SH(S)$ except that (1) spectra are replaced by chain complexes of $\Lambda$-modules and (2) the Nisnevich topology is replaced by the étale topology. So there's an obvious functor $SH(S) \to DA^{et}(S,\Lambda)$. There's an adic version of this when $\Lambda$ is the completion of a ring at an ideal.

On the other hand, if $D^{et}(S,\Lambda)$ denotes the derived category of sheaves of $\Lambda$-modules on the small étale site of $S$, there is also an obvious functor $D^{et}(S,\Lambda) \to DA^{et}(S,\Lambda)$. Ayoub's "rigidity theorem" (Theorem 4.1 in loc. cit.) combined with results of Gabber shows that this functor is an equivalence of categories if $S$ is excellent and if $\Lambda$ (or the quotient of $\Lambda$ by its ideal of definition) is a $\mathbb Z/n$-algebra for some $n$ invertible on $S$.

Under these assumptions, we therefore get a functor $$ SH(S) \to D^{et}(S,\Lambda). $$ It is symmetric monoidal and colimit-preserving, since the two functors considered above are.

This étale realization functor sends a smooth $S$-scheme $p\colon X\to S$ to $p_!p^!\Lambda$. An alternative approach to this functor would be to directly use the universal property of $SH(S)$ established by Robalo in K-theory and the bridge from motives to noncommutative motives (Corollary 2.39). With no assumptions on $S$, it is clear that the assignment $(p\colon X\to S)\mapsto p_!p^!\Lambda$ satisfies Nisnevich descent, homotopy invariance, and $\mathbb P^1$-stability, but promoting it to a symmetric monoidal functor is where the difficulty lies, and where excellence is needed.

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    $\begingroup$ I should add, it's only a technical matter to make $p\mapsto p_!p^!\Lambda$ into an (op?)lax symmetric monoidal functor, and it is known to be strict symmetric monoidal when $S$ is excellent, by work of Gabber. When $S$ is a field, however, this was proved by Deligne in SGA4½. $\endgroup$ – Marc Hoyois Nov 21 '15 at 2:19
  • $\begingroup$ Are you sure that Ayoub did not demand any finiteness of l-adic cohomological dimension restrictions? $\endgroup$ – Mikhail Bondarko Nov 21 '15 at 19:29
  • $\begingroup$ @MikhailBondarko If $S$ is excellent and $p$ is invertible on $S$, then for every strictly henselian local ring $A$ of $S$ and every $x\in Spec(A)$, $cd_p(\kappa(x))=dim(\overline{\{x\}})<\infty$. This finiteness condition, for all $p$ dividing $n$, is the assumption of Ayoub's rigidity theorem. $\endgroup$ – Marc Hoyois Nov 21 '15 at 20:08

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