Pro-algebraic versus continuous Galois cohomology, and schematic homotopy types

Question

I've been thinking about Bertrand Toen's approach to studying the homotopy theory of schemes, and I've come across an inconsistency in my understanding of the subject that I was hoping somebody might be able to iron out for me.

If I take a field $k$ of characteristic $\neq\ell$, then if I have understood this

http://www.aimath.org/WWN/motivesdessins/Toen.pdf

correctly, there should be an $\ell$-adic schematic homotopy type $h(X)$ associated to every smooth and projective variety over $k$, such that the 'absolute' $\ell$-adic cohomology of $X$, which coefficients in a local system, can be calculated as an appropriate hom set inside the derived category of perfect complexes on $h(X)$. Specifically, I understood that the derived category of perfect complexes on $h(X)$ should be equivalent to the full subcategory of the usual $\ell$-adic derived category $D^b_c(X_{\mathrm{et}},\mathbb{Q}_\ell)$ whose cohomology sheaves are lisse.

The reason this is confusing me is because I also thought that the schematic homotopy type associated to the point $\mathrm{Spec}(k)$ should be the classifying stack $BG_k^\mathrm{alg}$ associated to the $\mathbb{Q}_\ell$ pro-algebraic completion $G_k^\mathrm{alg}$ of the absolute Galois group of $k$.

But the derived category of perfect complexes $D_\mathrm{perf}(BG_k^\mathrm{alg})$ on this classifying stack, as far as I can tell, should not be equivalent to $D^b_c(\mathrm{Spec}(k)_\mathrm{et},\mathbb{Q}_\ell)$ - if I take some finite dimensional continuous $G_k$-representation $V$ then the cohomology computed in the former category will be the algebraic group cohomology $$\mathrm{Hom}_{D_\mathrm{perf}(BG_k^\mathrm{alg})}(\mathbb{Q}_\ell,V[i])=H^i(G_k^\mathrm{alg},V)$$ whereas the cohomology computed in the latter category will be continuous group cohomology $$\mathrm{Hom}_{D^b_\mathrm{lisse}(X_\mathrm{et},\mathbb{Q}_\ell)}(\mathbb{Q}_\ell,V[i])=H^i_\mathrm{cts}(G_k,V).$$ These two won't coincide, and this suggests that there are no suitable subcategories of $D_{\mathrm{perf}}(BG_k^\mathrm{alg})$ and $D^b_c(\mathrm{Spec}(k)_\mathrm{et},\mathbb{Q}_\ell)$ that will coincide.

So what's going on? I've convinced myself that the two expectations $$ D_\mathrm{perf}(h(X))\cong D^b_\mathrm{lisse}(X_\mathrm{et},\mathbb{Q}_\ell)$$ and $$h(\mathrm{Spec}(k))\cong BG_k^\mathrm{alg} $$ are incompatible - so which one should I hold on to? Or have I missed a trick somewhere?

EDIT: Clarified what I mean by the 'cohomology' calculated in both categories.

Answer 1

This answer is due to Jon Pridham.

While we might not expect $H^i(G_k,V)=H^i(G_k^\mathrm{alg},V)$ for every finite dimensional, continuous $G_k$-representation $V$, there are certain results from the motivic theory that suggest that this might be true if $k$ is a number field (or local field of char $0$), and that the representation 'comes from geometry'.

So although we possible should not expect a schematic homotopy type $h(\mathrm{Spec}(k))$ which is a gerbe, and whose derived category of perfect complexes is equivalent to $D^b_c(\mathrm{Spec}(k)_\mathrm{et},\mathbb{Q}_\ell)$, there may well be a gerbe whose derived category of perfect complexes is equivalent to the subcategory of $D^b_c(\mathrm{Spec}(k)_\mathrm{et},\mathbb{Q}_\ell)$ generated by complexes of geometric origin, at least when $k$ is a number field.

Essentially, the existence of the motivic $t$-structure 'almost' implies that the motivic schematic homotopy type is a gerbe, from which it would follow that the $\ell$-adic schematic homotopy type associated to motivic representations (i.e. representations of geometric origin) is a gerbe.