TITLE: Obstructions to descend Galois invariant cycles
QUESTION [16 upvotes]: Let $X$ be a smooth projective variety over $F$, and $E/F$ - finite Galois extension.
There is an extension of scalars map $CH^\*(X) \to CH^\*(X_E)$. The image lands in the Galois invariant part of $CH^\*(X_E)$, and in the case of rational coefficients, all Galois-invariant cycles are in the image (EDIT: this follows from taking the trace argument).
With integer coefficients Galois-invariant classes don't have to descend.
For example, for $CH^1(X) = Pic(X)$ there is an exact sequence:
$$
0 \to Pic(X) \to Pic(X_E)^{Gal(E/F)} \to Br(F),
$$
so we can say that the obstruction to descend a cycle lies in a Brauer group.
Are there any known obstructions to descend elements of higher groups $CH^i$ with integer coefficients?
In my case I have a cycle in $CH^*(X_E)^{Gal(E/F)}$ and I want to find out whether or not it's coming from $CH^*(X)$. 
(The actual cycle is described in here.)

REPLY [6 votes]: Dear Evgeny Shinder,
For  CH^2(X)  one can find an "obstruction" in a non-ramified cohomology. 
Let us write H^i_nr(X, \mu_l^j) for the intersection of kernels of the residu maps d_A in Galois cohomology, where A is running over all discrete valuation rings with field of fractions F(X) containing F. For example, if X is regular, one should take in account all valuations coming from codimension one points, but also the valuations coming from exceptional divisors of various blowing-ups.
If X is a smooth geometrically rational variety defined over a finite field F (i.e. rational over an algebraic closure \bar F of F), one can show that the cokernel of the map CH^2(X)->CH^2(X_{\bar F})^G is isomorphic, up to p-torsion (p=char F), to the group of non-ramified cohomology H^3_nr(X, Q/Z(2)). One can also find examples when this last group is non zero (see http://arxiv.org/abs/1004.1897).
Is it helpful for you?
Best regards,
Alena Pirutka.