The Stacks project

94.6 Finite étale covers

We define a category $\textit{FÉt}$ as follows:

  1. An object of $\textit{FÉt}$ is a finite étale morphism $Y \to X$ of schemes (by our conventions this means a finite étale morphism in $(\mathit{Sch}/S)_{fppf}$),

  2. A morphism $(b, a) : (Y \to X) \to (Y' \to X')$ of $\textit{FÉt}$ is a commutative diagram

    \[ \xymatrix{ Y \ar[d] \ar[r]_ b & Y' \ar[d] \\ X \ar[r]_ a & X' } \]

    in the category of schemes.

Thus $\textit{FÉt}$ is a category and

\[ p : \textit{FÉt} \to (\mathit{Sch}/S)_{fppf}, \quad (Y \to X) \mapsto X \]

is a functor. Note that the fibre category of $\textit{FÉt}$ over a scheme $X$ is just the category $\textit{FÉt}_ X$ studied in Fundamental Groups, Section 58.5.

Lemma 94.6.1. The functor

\[ p : \textit{FÉt} \longrightarrow (\mathit{Sch}/S)_{fppf} \]

defines a stack over $(\mathit{Sch}/S)_{fppf}$.

Proof. Fppf descent for finite étale morphisms follows from Descent, Lemmas 35.37.1, 35.23.23, and 35.23.29. Details omitted. $\square$


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