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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.

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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 $FinEt/X \simeq colim_\alpha FinEt/X_\alpha$.

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