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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.

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  • $\begingroup$ Sorry, which particular section of the thesis did you have in mind? It seems to be talking about pro-finite homotopy types to me... $\endgroup$ – ChrisLazda Nov 6 '13 at 17:22
  • $\begingroup$ Oh, I see. My apologies. It seems the schematic homotopy type is not the profinite homotopy type (I thought they were two names for the same thing). $\endgroup$ – Qiaochu Yuan Nov 6 '13 at 18:27
  • $\begingroup$ general triangulated category comment: could it be that you're taking cohomology with respect to two different t-structures? (so that the equivalence holds, but without taking one standard heart in the other) $\endgroup$ – John Salvatierrez Nov 7 '13 at 0:03
  • $\begingroup$ No, I don't think so. By cohomology, I really mean $Hom_{D}(1,V[i])$, where $D$ is either of the triangulated categories, so $t$-structures don't come into it at all. I'll edit the question to make this clearer. $\endgroup$ – ChrisLazda Nov 7 '13 at 8:30
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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.

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