Lemma 85.31.2. Let $X \to S$ be a morphism of schemes. Suppose $Y \to (X/S)_\bullet $ is a cartesian morphism of simplicial schemes. Let $y \in Y_0$ be a point. If $X \to S$ is quasi-compact, then

is a quasi-compact subset of $Y_0$.

Lemma 85.31.2. Let $X \to S$ be a morphism of schemes. Suppose $Y \to (X/S)_\bullet $ is a cartesian morphism of simplicial schemes. Let $y \in Y_0$ be a point. If $X \to S$ is quasi-compact, then

\[ T_ y = \{ y' \in Y_0 \mid \exists \ y_1 \in Y_1: d^1_1(y_1) = y, d^1_0(y_1) = y'\} \]

is a quasi-compact subset of $Y_0$.

**Proof.**
Let $F_ y$ be the scheme theoretic fibre of $d^1_1 : Y_1 \to Y_0$ at $y$. Then we see that $T_ y$ is the image of the morphism

\[ \xymatrix{ F_ y \ar[r] \ar[d] & Y_1 \ar[r]^{d^1_0} \ar[d]^{d^1_1} & Y_0 \\ y \ar[r] & Y_0 & } \]

Note that $F_ y$ is quasi-compact. This proves the lemma. $\square$

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