Let $\mathcal{T}$ be a triangulated category which has arbitraty direct sums. An object $E\in \mathcal{T}$ is called compact if the functor Hom$(E,-)$ commutes with arbitrary direct sums. A compact object $E$ is called a compact generator of $\mathcal{T}$ if for any object $N$, $\text{Hom}(E,N[i])=0$, $\forall i\in \mathbb{Z}$ implies $N=0$.
Now consider a quasi-compact, separated scheme $X$ over a base field $k$. Let $D(X)$ be the derived category of quasi-coherent $\mathcal{O}_X$-modules and $E$ be a compact generator of $D(X)$.
Let $l/k$ be a field extension and $X_l$ be the base change sheme with $p: X_l \to X$ the natural projection. Let $E_l:=p^*E$. We know $E$ must be a perfect complex hence so is $E_l$. Therefore $E_l$ is a compact object in $D(X_l)$.
My question is: is it true that $E_l$ is also a compact generator of $D(X_l)$? We know that if $l/k$ is a finite extension then it is true. In fact this is a special case of BoVdB Lemma 3.4.1. But what about $l/k$ is not a finite extension?
