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!
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
Sign up to join this communityLet $k$ 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!
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.