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 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.

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.

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 some 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 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.

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 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.