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

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.

• I recommend in this context the recent paper of Clausen and Mathew arxiv.org/abs/1905.06611 - this has some cleaner and more general results than the original result by Thomason. (Note that $p$-completed $TC$ of a ring vanishes if $p$ is invertible in the ring.) – Lennart Meier Aug 18 '19 at 1:36
• thanks lennart! I actually did take a look at that paper and I think resolved my issue (see the answer below), though I cited the étale-descent of localized algebraic K-theory to thomason rather than clausen-mathew for reasons of priority. – xir Aug 18 '19 at 1:38

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 $$\text{Map}_\Gamma(X^\theta, \mathcal{\widehat{K}})$$.