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Tag 0AP4

Chapter 34: Descent > Section 34.32: Fully faithfulness of the pullback functors

Lemma 34.32.13. Let $X \to S$ be a surjective, quasi-compact, flat morphism of schemes. Let $(V, \varphi)$ be a descent datum relative to $X/S$. Suppose that for all $v \in V$ there exists an open subscheme $v \in W \subset V$ such that $\varphi(W \times_S X) \subset X \times_S W$ and such that the descent datum $(W, \varphi|_{W \times_S X})$ is effective. Then $(V, \varphi)$ is effective.

Proof. Let $V = \bigcup W_i$ be an open covering with $\varphi(W_i \times_S X) \subset X \times_S W_i$ and such that the descent datum $(W_i, \varphi|_{W_i \times_S X})$ is effective. Let $U_i \to S$ be a scheme and let $\alpha_i : (X \times_S U_i, can) \to (W_i, \varphi|_{W_i \times_S X})$ be an isomorphism of descent data. For each pair of indices $(i, j)$ consider the open $\alpha_i^{-1}(W_i \cap W_j) \subset X \times_S U_i$. Because everything is compatible with descent data and since $\{X \to S\}$ is an fpqc covering, we may apply Lemma 34.10.2 to find an open $V_{ij} \subset V_j$ such that $\alpha_i^{-1}(W_i \cap W_j) = X \times_S V_{ij}$. Now the identity morphism on $W_i \cap W_j$ is compatible with descent data, hence comes from a unique morphism $\varphi_{ij} : U_{ij} \to U_{ji}$ over $S$ (see Remark 34.32.12). Then $(U_i, U_{ij}, \varphi_{ij})$ is a glueing data as in Schemes, Section 25.14 (proof omitted). Thus we may assume there is a scheme $U$ over $S$ such that $U_i \subset U$ is open, $U_{ij} = U_i \cap U_j$ and $\varphi_{ij} = \text{id}_{U_i \cap U_j}$, see Schemes, Lemma 25.14.1. Pulling back to $X$ we can use the $\alpha_i$ to get the desired isomorphism $\alpha : X \times_S U \to V$. $\square$

    The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 7948–7956 (see updates for more information).

    \begin{lemma}
    \label{lemma-effective-for-fpqc-is-local-upstairs}
    Let $X \to S$ be a surjective, quasi-compact, flat morphism of
    schemes. Let $(V, \varphi)$ be a descent datum relative to $X/S$.
    Suppose that for all $v \in V$ there exists an open subscheme
    $v \in W \subset V$ such that $\varphi(W \times_S X) \subset X \times_S W$
    and such that the descent datum $(W, \varphi|_{W \times_S X})$
    is effective. Then $(V, \varphi)$ is effective.
    \end{lemma}
    
    \begin{proof}
    Let $V = \bigcup W_i$ be an open covering with
    $\varphi(W_i \times_S X) \subset X \times_S W_i$
    and such that the descent datum $(W_i, \varphi|_{W_i \times_S X})$
    is effective. Let $U_i \to S$ be a scheme and let
    $\alpha_i : (X \times_S U_i, can) \to (W_i, \varphi|_{W_i \times_S X})$
    be an isomorphism of descent data. For each pair of indices
    $(i, j)$ consider the open
    $\alpha_i^{-1}(W_i \cap W_j) \subset X \times_S U_i$.
    Because everything is compatible with descent data
    and since $\{X \to S\}$ is an fpqc covering, we
    may apply Lemma \ref{lemma-open-fpqc-covering}
    to find an open $V_{ij} \subset V_j$ such that
    $\alpha_i^{-1}(W_i \cap W_j) = X \times_S V_{ij}$.
    Now the identity morphism on $W_i \cap W_j$ is
    compatible with descent data, hence comes from a
    unique morphism $\varphi_{ij} : U_{ij} \to U_{ji}$ over $S$
    (see Remark \ref{remark-morphisms-of-schemes-satisfy-fpqc-descent}).
    Then $(U_i, U_{ij}, \varphi_{ij})$ is a glueing
    data as in Schemes, Section \ref{schemes-section-glueing-schemes}
    (proof omitted). Thus we may assume there is a scheme $U$ over $S$
    such that $U_i \subset U$ is open, $U_{ij} = U_i \cap U_j$ and
    $\varphi_{ij} = \text{id}_{U_i \cap U_j}$, see
    Schemes, Lemma \ref{schemes-lemma-glue}.
    Pulling back to $X$ we can use the $\alpha_i$ to
    get the desired isomorphism $\alpha : X \times_S U \to V$.
    \end{proof}

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