ag.algebraic geometry - Is the scalar extension functor for Chow motives conservative?

With rational coefficients, the answer is yes.

The first case to understand is when $E$ is a finite algebraic extension of $F$.
In the case when moreover $E$ is purely inseparable, then the extension of scalars
functors $CHM(F)to CHM(E)$ is fully faithful, and if $E$ is Galois of degree $d$, then the extension of scalars functor

$$pi^star:CHM(F)to CHM(E)$$
has a right adjoint $pi_star$, and for any motive $M$ over $F$, there is a trace map
$$tr_M : pi_star pi^star(M)to M$$
whose composition with the unit map
$$Mto pi_star pi^star(M)$$
is multiplication by $d$. If you work with rational coefficients, this implies that $pi^star$ is then conservative and faithful.

From there, to prove the general case, we may assume that
$E$ is a filtered colimit of smooth $F$-algebras $A_i$.
But then, for any index $i$, possibly after taking a finite extension of $F$,
the map $Fto A_i$ has a retraction, so that, writing
$CHM(E)$ as the $2$-colimit of the categories $CHM(A_i)$, we see easily that the extension of scalars functors is again faithful and conservative (for Chow motives over a smooth $F$-algebra, see for instance Definition 5.16 in Levine's paper arXiv:0807.2265).

If you really want Chow motives with integral coefficients, you mays still have to invert the (exponential) characteristic of $F$. Then, assuming furthermore that $F$ and $E$ are algebraically closed, the extension of scalars functors will be conservative again (this uses rigidity theorems; see O. Röndigs and P. A. Østvær, Rigidity in motivic homotopy theory, Math. Ann. 341 (2008), 651-675).

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