Lemma 35.35.11. Let $S$ be a scheme. Let $\mathcal{U} = \{ U_ i \to S\} _{i \in I}$, and $\mathcal{V} = \{ V_ j \to S\} _{j \in J}$, be fpqc-coverings of $S$. If $\mathcal{U}$ is a refinement of $\mathcal{V}$, then the pullback functor
\[ \text{descent data relative to } \mathcal{V} \longrightarrow \text{descent data relative to } \mathcal{U} \]
is fully faithful. In particular, the category of schemes over $S$ is identified with a full subcategory of the category of descent data relative to any fpqc-covering of $S$.
Proof.
Consider the fpqc-covering $\mathcal{W} = \{ U_ i \times _ S V_ j \to S\} _{(i, j) \in I \times J}$ of $S$. It is a refinement of both $\mathcal{U}$ and $\mathcal{V}$. Hence we have a $2$-commutative diagram of functors and categories
\[ \xymatrix{ DD(\mathcal{V}) \ar[rd] \ar[rr] & & DD(\mathcal{U}) \ar[ld] \\ & DD(\mathcal{W}) & } \]
Notation as in the proof of Lemma 35.35.9 and commutativity by Lemma 35.34.8 part (3). Hence clearly it suffices to prove the functors $DD(\mathcal{V}) \to DD(\mathcal{W})$ and $DD(\mathcal{U}) \to DD(\mathcal{W})$ are fully faithful. This follows from Lemma 35.35.9 as desired.
$\square$
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Comment #503 by Kestutis Cesnavicius on
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