Question about an implication of Thomason's étale descent spectral sequence

Question

On page 5 of this paper by Dwyer and Mitchell, it is said that Thomason's étale descent spectral sequence from his paper Algebraic K-theory and étale cohomology, which reads

$$H^p_{\acute{e}t}(X, \mathbb{Z}_l(-q/2)) \Rightarrow \pi_{-q-p}\hat{L}KX$$

where $KX$ is the algebraic K-theory spectrum of $X$ and $\hat{L}$ denotes the $\ell$-completed Bousefield localization at topological K-theory, implies the natural isomorphism

$$\pi_i\widehat{L}(KR) \cong \pi_i\text{Map}_{\Gamma_F'}(X_+^\theta, \hat{\mathcal{K}})$$

where here, $R$ is the ring of integers localized away from $\ell$ of some totally real field $F$, $X$ is a space realizing the étale homotopy type of $R$, $\theta$ is the character of $\pi_1(X)$ corresponding to the $\ell$-adic cyclotomic character, $X^\theta$ is the cover of $X$ corresponding to the kernel of $\theta$, with the corresponding action of $\theta$ of the Galois/fundamental group. The subscript plus, as usual, denotes the unreduced suspension spectrum. The Galois group of the $\ell$-adic cyclotomic extension is in particular $\Gamma_F'$, and lives inside $\Gamma'\cong \mathbb{Z}_l^\times$ (i.e. the corresponding Galois group over $\mathbb{Q}$.)

The $\text{Map}$ is then an equivariant mapping spectrum, with the action of $\Gamma'$ via Adams operations on the target.

(It is assumed we have a rigid, i.e. not just up to homotopy, action of $\Gamma'$ on $\hat{\mathcal{K}}$.)

So, how does this follow? There is no real explanation given, which makes me think it has to be simple. But I'm confused what the relationship of the localized algebraic K-theory groups to étale cohomology has to do with this twisted equivariant mapping spectrum.

Apologies if this is some very formal thing; I'm pretty un-versed in these matters. I'll delete this question if it turns out to just be very obvious somehow.

Answer 1

I think I understand the situation a little better now. Thomason's descent spectral sequence requires the base scheme to be ``$\ell$-good'', or have all the $\ell$-power roots of unity, since the point is that what is proven is that we have a weak equivalence $\hat{L}KS \simeq \hat{L}\mathbb{H}_{\acute{e}t}(S,K)$, where the latter is the localized version of Thomason's hypercohomology spectrum, with coefficients in the sheaf of algebraic K-theory spectra. This his way of constructing the upgraded version of the usual descent spectral sequences for sheaves taking values in the usual modules.

The point is that if $X$ is $\ell$-good, and a few other nice technical conditions, the étale-local stalks of the sheaf $K$ are the K-theory spectra of strict Henselian rings, so their homotopy groups agree with the algebraic K-theory of their (separably closed) residue fields, hence (by a theorem of Suslin) are locally $\mathbb{Z}_l(n)$ in degree $n$ after being hit by $\hat{L}$. (I'm not super clear why we can pass the $\hat{L}$ through the hypercohomology spectrum functor; after all, localization at $K$-theory is smashing, but $\ell$-completion is not. Maybe just passing the K-theory localization through is enough if we take periodic $K$-theory, and then we can just $l$-complete at the end?)

Since $X$ globally contains all the $\ell$-power roots of unity by the $\ell$-good hypothesis, the localized sheaf of homotopy groups in degree $n$ is actually the constant sheaf $\mathbb{Z}_l(n)$; said differently, the sheaf of spectra is weakly equivalent to the constant sheaf $\widehat{\mathcal{K}}$ of $\mathcal{l}$-completed (topological, complex) K-theory spectra. So then by applying the same descent argument using the complete topological K-theory functor, we also get a weak equivalence with $\hat{L}\text{Map}(X,\hat{\mathcal{K}})=\text{Map}(X,\hat{\mathcal{K}})$.

So, returning to the situation at hand, if $\text{Spec }R$ is just the $\ell$-inverted ring of integers of a totally real field with étale homotopy type $X$, the thing we want to do is apply Thomason's construction to the $\ell$-good cyclotomic extension $R_\infty$, whose étale homotopy type is exactly $X^\theta$. Then we find that $\hat{L}KR_\infty \simeq \hat{L}\mathbb{H}_{\acute{e}t}(R_\infty,\widehat{\mathcal{K}})\simeq \mathcal{\widehat{K}}(X^\theta)$. Étale descending, the homotopy fixed points of the LHS is our $\hat{L}KR$. (after Thomason, $K(1)$-localized algebraic K-theory does satisfy étale descent, unlike the original.)

On the RHS, to compute the homotopy fixed points, we can compute that the $\Gamma$-action on the target $\mathcal{\widehat{K}}$ is by the Adams operations, remembering that it comes originally from algebraic K-theory via Suslin's theorem. (For example, in our case, since all the residue fields are algebraic over a finite field, this is just Quillen's computation.) Hence the descended version is just precisely our equivariant mapping space $\text{Map}_\Gamma(X^\theta, \mathcal{\widehat{K}})$.

I'm definitely not an expert on these matters, so please point out if there are glaring errors or missing pieces!