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
Comments (1)
Comment #503 by Kestutis Cesnavicius on
There are also: