Let $X$ be a smooth projective variety over a field $k$ and $D^b(X)$ its bounded derived category. Let $\bar{X}$ the base change to $\bar{k}$. Let $A$ be a triangulated subcategory of $D^b(X)$ that $\bar{A} $ is admissible inside $D^b(\bar{X})$ over $\bar{k}$. Is $A$ admissible also inside $D^b(X)$ over $k$? (admissible means that the embedding functor admits both left and right adjoint functors)



admissible subcategories over non algebraically closed fields
I think the answer should be yes, but the statement should be more accurate. First, in the definition of $\bar{A}$ you first extend scalars and then take the triangulated hull and add all direct summands. consequently, if you want the statement to be true you should add all direct summands to $A$ as well (or assume that $A$ is Karoubian complete from the start). 


The proof should go as follows. Assume that $D(\bar{X}) = \langle \bar{A}, \bar{B} \rangle$ be a semiorthogonal decomposition. First one should check that it is invariant under the Galois action. Then one should check that an object in $D(\bar{X})$ is Galois invariant if and only if it is in $D(X)$. Then one should restrict the semiorthogonal decomposition of $D(X)$ by intersecting the above decomposition with $D(X)$.


Of course, there are a lot of subtle points, for example the extension of scalars is not fully faithful, but I do believe that with a bit of accuracy one can do this.


