The Stacks project

Example 92.9.1 (Schemes). Let $\mathcal{F}$ be the category defined as follows

  1. an object is a pair $(A, X)$ consisting of an object $A$ of $\mathcal{C}_\Lambda $ and a scheme $X$ flat over $A$, and

  2. a morphism $(f, g) : (B, Y) \to (A, X)$ consists of a morphism $f : B \to A$ in $\mathcal{C}_\Lambda $ together with a morphism $g : X \to Y$ such that

    \[ \xymatrix{ X \ar[r]_ g \ar[d] & Y \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r]^ f & \mathop{\mathrm{Spec}}(B) } \]

    is a cartesian commutative diagram of schemes.

The functor $p : \mathcal{F} \to \mathcal{C}_\Lambda $ sends $(A, X)$ to $A$ and $(f, g)$ to $f$. It is clear that $p$ is cofibred in groupoids. Given a scheme $X$ over $k$, let $x_0 = (k, X)$ be the corresponding object of $\mathcal{F}(k)$. We set

\[ \mathcal{D}\! \mathit{ef}_ X = \mathcal{F}_{x_0} \]

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