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57.5 Finite étale morphisms
In this section we prove enough basic results on finite étale morphisms to be able to construct the étale fundamental group.
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Let $X$ be a scheme. We will use the notation $textit{FÉt}_ X$ to denote the category of schemes finite and étale over $X$. Thus
an object of $textit{FÉt}_ X$ is a finite étale morphism $Y to X$ with target $X$, and
a morphism in $textit{FÉt}_ X$ from $Y to X$ to $Y' to X$ is a morphism $Y to Y'$ making the diagram
[ xymatrix{ Y ar[rr] ar[rd] & & Y' ar[ld] & X } ]commute.
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We will often call an object of $textit{FÉt}_ X$ a finite étale cover of $X$ (even if $Y$ is empty). It turns out that there is a stack $p : textit{FÉt} to mathit{Sch}$ over the category of schemes whose fibre over $X$ is the category $textit{FÉt}_ X$ just defined. See Examples of Stacks, Section 93.6.
Let $X$ be a scheme. A geometric point of $X$ is a morphism $mathop{mathrm{Spec}}(k) to X$ where $k$ is algebraically closed. Such a point is usually denoted $overline{x}$, i.e., by an overlined small case letter. We often use $overline{x}$ to denote the scheme $mathop{mathrm{Spec}}(k)$ as well as the morphism, and we use $kappa (overline{x})$ to denote $k$. We say $overline{x}$ lies over $x$ to indicate that $x in X$ is the image of $overline{x}$. We will discuss this further in Étale Cohomology, Section 58.29. Given $overline{x}$ and an étale morphism $U to X$ we can consider
Since $U_{overline{x}}$ as a scheme over $overline{x}$ is a disjoint union of copies of $overline{x}$ (Morphisms, Lemma 29.36.7) we can also describe this set as
The assignment $U mapsto |U_{overline{x}}|$ is a functor which is often denoted $F_{overline{x}}$.
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