Realization Functor From $SH$ to Derived Category of $Gal$-Modules

Question

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!

Answer 1

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.