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Let $X$ be a (sufficiently nice) algebraic space, one can define the category of étale motives $\mathbf{DA}(X,R)$ (with $R$ a ring) like the case of schemes (see for instance, La réalisation étale et les opérations de Grothendieck by J. Ayoub).

In loc.cit. Theorem 3.9, it is proved that if $X$ is a scheme, then the $2$-functor $\mathbf{DA}(-,R)$ is separated, meaning that if $f \colon X \longrightarrow Y$ is surjective then $f^* \colon \mathbf{DA}(Y,R) \longrightarrow \mathbf{DA}(X,R)$ is conservative.

If $R$ is $\mathbb{Q}$-algebra, then $\mathbf{DA}(X,R)$ is $\mathbb{Q}$-linear and separated and therefore satisfies étale descent by Proposition 3.3.33 in Triangulated Categories of Mixed Motives by Cisinski and Déglise. In particular, take soming first terms of the Cech étale hypercover one gets an exact sequence $$0 \rightarrow \mathbf{DA}(X) \rightarrow \mathbf{DA}(U) \rightarrow \mathbf{DA}(U \times_X U)$$ with $U \longrightarrow X$ an étale, surjective morphism.

My question. Do we have the same sequence as above with $U$, $X$ algebraic spaces?

After some few attempts, I can derive the exactness at $\mathbf{DA}(X)$ but could not do anything with the next exactness or derive it from the case of schemes.

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    $\begingroup$ $X\mapsto \mathbf{DA}(X,R)$ is an étale sheaf of $\infty$-categories, more or less by construction. $R$ doesn't have to be a $\mathbb Q$-algebra. $\endgroup$
    – Marc Hoyois
    Commented Oct 3 at 6:51
  • $\begingroup$ @MarcHoyois do you have a precise reference? $\endgroup$
    – Alexey Do
    Commented Oct 5 at 7:37
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    $\begingroup$ No, but it's almost formal. The key point is that, if $Y$ is étale over $X$, then $Shv(Sm_Y) = Shv(Sm_X)_{/Y}$ and the latter is a sheaf in $Y$ by topos theory (Theorem 6.1.3.9(3) in Higher Topos Theory). $\endgroup$
    – Marc Hoyois
    Commented Oct 5 at 15:53
  • $\begingroup$ So this means the descent already holds at the level of sheaves $Shv(-)$? Since I am not really good at $\infty$-categories, I think at the level of ordinary categories: let $pr_1,pr_2: U \times_X U \to U$ be projections, a sheaf $F \in Shv(Sm/U)$ so that $(pr_1)^*(F)$ weakly equivalent to $(pr_2)^*(F)$ can be expressed (up to weak equivalence) as a homotopy equaliser of the form $u_*(F) \rightarrow u_*(pr_1)_*(pr_1)^*(F)$. $\endgroup$
    – Alexey Do
    Commented Oct 6 at 7:13
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    $\begingroup$ Yes, $X\mapsto Shv(Sm_X)$ is a sheaf of $\infty$-categories, and the steps to get from there to $DA$ (e.g. passing to $\mathbb P^1$-spectra) preserve limits and hence sheaves. However, descent involves simplicial limits, not equalizers, so your description of descent is incorrect. I strongly suggest getting more familiar with higher categories if you want to work with derived categories of any kind; you will be severely restricted without them. $\endgroup$
    – Marc Hoyois
    Commented Oct 6 at 12:02

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