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I have found two definitions of geometric fiber. Let $f:X\rightarrow Y$ be a morphism of schemes and $y \in Y$ then the geometric fiber over $y$ is

$$X_y\times_{Speck(y)}Spec K $$

where $X_y=X\times_Y Spec k(y)$ is the fiber over $y$ and in some book they take $K$ as the algebraic closure but in other books as the separable closure.

What is the difference between both definitions?

Thanks a lot.

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  • $\begingroup$ What is the actual question? For some purposes the difference is irrelevant, while for others the difference is crucial. It depends on what you want to do. $\endgroup$
    – Kestutis Cesnavicius
    Commented Jan 12, 2014 at 19:51
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    $\begingroup$ The geometry and etale sites are the same, but reducedness is a big distinction. One merit of just going up to the separable closure is that you can often use Galois descent to return to the "actual fiber", which fails completely if you go up to the algebraic closure (when $k(y)$ is not perfect). On the other hand, if you want to get a rational point you might need to make an inseparable extension on $k=k(y)$. But such extension can fail to commute with the formation of nilradical or "$k$-unipotent radical", etc. Kestutis' final sentence hits the nail on the head. $\endgroup$
    – user76758
    Commented Jan 12, 2014 at 20:52
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    $\begingroup$ @user45442: Here's an example illustrating the need for context. For an affine finite type group scheme $G$ over a field $k$, $(G_{\overline{k}})_{\rm{red}}$ is a smooth subgroup scheme of $G_{\overline{k}}$ (usually not normal!) but $(G_{k_s})_{\rm{red}} = (G_{\rm{red}})_{k_s}$ is generally not a subgroup scheme of $G_{k_s}$. Yet for some big $n$ the quotient $G/\ker(F_{G/k,n})$ modulo kernel of the $n$-fold relative Frobenius morphism is smooth, so if quotient by an infinitesimal normal $k$-subgroup is harmless for some purpose then one can pass to smooth groups over $k$ after all... $\endgroup$
    – user76758
    Commented Jan 12, 2014 at 22:59
  • $\begingroup$ @KestutisCesnavicius I'm studing étale morphism, and I found those two definitions and I was worry about id there is a big difference. $\endgroup$
    – user45442
    Commented Jan 14, 2014 at 0:35
  • $\begingroup$ @user76758 Thanks for point me out that it would be important for use Galois descent. $\endgroup$
    – user45442
    Commented Jan 14, 2014 at 0:39

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