Let $k'/k$ be an extension of algebraically closed fields of characteristic $0$, and $X$ a concentrated (i.e. quasi-compact and quasi-separated) scheme over $k$.


**Question**: is the pullback functor from finite étale covers of $X$ to finite étale covers of $X\_{k'}$ an equivalence?


If $X$ is connected, then by SGA1, Exposé XIII, Proposition 4.6b we know that $\pi\_1(X)=\pi\_1(X\_{k'})$, and hence the answer is yes. If the connected components of $X$ are open (for example if $X$ is of finite type), then we can reduce ourselves to the connected case, but in general connected components are not open.



Finite étale covers of concentrated schemes and extension of base field
Yes, you can reduce to the finite type case by noetherian approximation (Appendix C in Thomason-Trobaugh). Namely, you can write $X=lim\_\alpha X\_\alpha$ where $X\_\alpha$ is of finite type over $k$. Then there is an equivalence of categories $FinEt/X \simeq colim\_\alpha FinEt/X\_\alpha$.


