The Stacks project

Lemma 85.27.4. Let $f : X \to S$ be a morphism of schemes. Let $\pi : Y \to (X/S)_\bullet $ be a cartesian morphism of simplicial schemes. Set $V = Y_0$ considered as a scheme over $X$. The morphisms $d^1_0, d^1_1 : Y_1 \to Y_0$ and the morphism $\pi _1 : Y_1 \to X \times _ S X$ induce isomorphisms

\[ \xymatrix{ V \times _ S X & & Y_1 \ar[ll]_-{(d^1_1, \text{pr}_1 \circ \pi _1)} \ar[rr]^-{(\text{pr}_0 \circ \pi _1, d^1_0)} & & X \times _ S V. } \]

Denote $\varphi : V \times _ S X \to X \times _ S V$ the resulting isomorphism. Then the pair $(V, \varphi )$ is a descent datum relative to $X \to S$.

Proof. This is a special case of (part of) Lemma 85.27.2 as the displayed equation of that lemma is equivalent to the cocycle condition of Descent, Definition 35.34.1. $\square$

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