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Tag 02W3

Chapter 34: Descent > Section 34.33: Descending types of morphisms

Lemma 34.33.2. Let $\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\tau \in \{fpqc, fppf, {\acute{e}tale}, smooth, syntomic\}$. Suppose that

  1. $\mathcal{P}$ is stable under any base change (see Schemes, Definition 25.18.3), and
  2. for any surjective morphism of affines $X \to S$ which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether $\tau$ is fpqc, fppf, étale, smooth, or syntomic, any descent datum $(V, \varphi)$ relative to $X$ over $S$ such that $\mathcal{P}$ holds for $V \to X$ is effective.

Then morphisms of type $\mathcal{P}$ satisfy descent for $\tau$-coverings.

Proof. Let $S$ be a scheme. Let $\mathcal{U} = \{\varphi_i : U_i \to S\}_{i \in I}$ be a $\tau$-covering of $S$. Let $(X_i, \varphi_{ii'})$ be a descent datum relative to $\mathcal{U}$ and assume that each morphism $X_i \to U_i$ has property $\mathcal{P}$. We have to show there exists a scheme $X \to S$ such that $(X_i, \varphi_{ii'}) \cong (U_i \times_S X, can)$.

Before we start the proof proper we remark that for any family of morphisms $\mathcal{V} : \{V_j \to S\}$ and any morphism of families $\mathcal{V} \to \mathcal{U}$, if we pullback the descent datum $(X_i, \varphi_{ii'})$ to a descent datum $(Y_j, \varphi_{jj'})$ over $\mathcal{V}$, then each of the morphisms $Y_j \to V_j$ has property $\mathcal{P}$ also. This is true because we assumed that $\mathcal{P}$ is stable under any base change and the definition of pullback (see Definition 34.31.9). We will use this without further mention.

First, let us prove the lemma when $S$ is affine. By Topologies, Lemma 33.9.8, 33.7.4, 33.4.4, 33.5.4, or 33.6.4 there exists a standard $\tau$-covering $\mathcal{V} : \{V_j \to S\}_{j = 1, \ldots, m}$ which refines $\mathcal{U}$. The pullback functor $DD(\mathcal{U}) \to DD(\mathcal{V})$ between categories of descent data is fully faithful by Lemma 34.32.11. Hence it suffices to prove that the descent datum over the standard $\tau$-covering $\mathcal{V}$ is effective. By Lemma 34.31.5 this reduces to the covering $\{\coprod_{j = 1, \ldots, m} V_j \to S\}$ for which we have assumed the result in property (2) of the lemma. Hence the lemma holds when $S$ is affine.

Assume $S$ is general. Let $V \subset S$ be an affine open. By the properties of site the family $\mathcal{U}_V = \{V \times_S U_i \to V\}_{i \in I}$ is a $\tau$-covering of $V$. Denote $(X_i, \varphi_{ii'})_V$ the restriction (or pullback) of the given descent datum to $\mathcal{U}_V$. Hence by what we just saw we obtain a scheme $X_V$ over $V$ whose canonical descent datum with respect to $\mathcal{U}_V$ is isomorphic to $(X_i, \varphi_{ii'})_V$. Suppose that $V' \subset V$ is an affine open of $V$. Then both $X_{V'}$ and $V' \times_V X_V$ have canonical descent data isomorphic to $(X_i, \varphi_{ii'})_{V'}$. Hence, by Lemma 34.32.11 again we obtain a canonical morphism $\rho^V_{V'} : X_{V'} \to X_V$ over $S$ which identifies $X_{V'}$ with the inverse image of $V'$ in $X_V$. We omit the verification that given affine opens $V'' \subset V' \subset V$ of $S$ we have $\rho^V_{V''} = \rho^V_{V'} \circ \rho^{V'}_{V''}$.

By Constructions, Lemma 26.2.1 the data $(X_V, \rho^V_{V'})$ glue to a scheme $X \to S$. Moreover, we are given isomorphisms $V \times_S X \to X_V$ which recover the maps $\rho^V_{V'}$. Unwinding the construction of the schemes $X_V$ we obtain isomorphisms $$ V \times_S U_i \times_S X \longrightarrow V \times_S X_i $$ compatible with the maps $\varphi_{ii'}$ and compatible with restricting to smaller affine opens in $X$. This implies that the canonical descent datum on $U_i \times_S X$ is isomorphic to the given descent datum and we win. $\square$

    The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 8048–8066 (see updates for more information).

    \begin{lemma}
    \label{lemma-descending-types-morphisms}
    Let $\mathcal{P}$ be a property of morphisms of schemes over a base.
    Let $\tau \in \{fpqc, fppf, \etale, smooth, syntomic\}$.
    Suppose that
    \begin{enumerate}
    \item $\mathcal{P}$ is stable under any base change
    (see Schemes, Definition \ref{schemes-definition-preserved-by-base-change}),
    and
    \item for any surjective morphism of affines
    $X \to S$ which is flat, flat of finite presentation,
    \'etale, smooth or syntomic depending on whether $\tau$ is
    fpqc, fppf, \'etale, smooth, or syntomic,
    any descent datum $(V, \varphi)$ relative
    to $X$ over $S$ such that $\mathcal{P}$ holds for
    $V \to X$ is effective.
    \end{enumerate}
    Then morphisms of type $\mathcal{P}$ satisfy descent for $\tau$-coverings.
    \end{lemma}
    
    \begin{proof}
    Let $S$ be a scheme.
    Let $\mathcal{U} = \{\varphi_i : U_i \to S\}_{i \in I}$
    be a $\tau$-covering of $S$.
    Let $(X_i, \varphi_{ii'})$ be a descent datum relative to
    $\mathcal{U}$ and assume that each morphism $X_i \to U_i$ has property
    $\mathcal{P}$. We have to show there exists a scheme $X \to S$ such that
    $(X_i, \varphi_{ii'}) \cong (U_i \times_S X, can)$.
    
    \medskip\noindent
    Before we start the proof proper we remark that for any
    family of morphisms $\mathcal{V} : \{V_j \to S\}$ and any
    morphism of families $\mathcal{V} \to \mathcal{U}$, if we pullback
    the descent datum $(X_i, \varphi_{ii'})$ to a descent datum
    $(Y_j, \varphi_{jj'})$ over $\mathcal{V}$, then each of the
    morphisms $Y_j \to V_j$ has property $\mathcal{P}$ also.
    This is true because we assumed that $\mathcal{P}$ is stable
    under any base change and the definition of pullback
    (see Definition \ref{definition-pullback-functor-family}).
    We will use this without further mention.
    
    \medskip\noindent
    First, let us prove the lemma when $S$ is affine.
    By Topologies, Lemma
    \ref{topologies-lemma-fpqc-affine},
    \ref{topologies-lemma-fppf-affine},
    \ref{topologies-lemma-etale-affine},
    \ref{topologies-lemma-smooth-affine}, or
    \ref{topologies-lemma-syntomic-affine}
    there exists a standard $\tau$-covering
    $\mathcal{V} : \{V_j \to S\}_{j = 1, \ldots, m}$
    which refines $\mathcal{U}$. The pullback functor
    $DD(\mathcal{U}) \to DD(\mathcal{V})$
    between categories of descent data is fully faithful
    by Lemma \ref{lemma-refine-coverings-fully-faithful}.
    Hence it suffices to prove that the descent datum over
    the standard $\tau$-covering $\mathcal{V}$ is effective.
    By Lemma \ref{lemma-family-is-one} this reduces to the covering
    $\{\coprod_{j = 1, \ldots, m} V_j \to S\}$ for which we have
    assumed the result in property (2) of the lemma.
    Hence the lemma holds when $S$ is affine.
    
    \medskip\noindent
    Assume $S$ is general. Let $V \subset S$ be an affine open.
    By the properties of site the family
    $\mathcal{U}_V = \{V \times_S U_i \to V\}_{i \in I}$ is a
    $\tau$-covering of $V$. Denote
    $(X_i, \varphi_{ii'})_V$ the restriction (or pullback) of
    the given descent datum to $\mathcal{U}_V$.
    Hence by what we just saw we obtain a scheme $X_V$ over $V$
    whose canonical descent datum with respect to
    $\mathcal{U}_V$ is isomorphic to $(X_i, \varphi_{ii'})_V$.
    Suppose that $V' \subset V$ is an affine open of $V$.
    Then both $X_{V'}$ and $V' \times_V X_V$ have canonical
    descent data isomorphic to $(X_i, \varphi_{ii'})_{V'}$.
    Hence, by Lemma \ref{lemma-refine-coverings-fully-faithful}
    again we obtain a canonical morphism
    $\rho^V_{V'} : X_{V'} \to X_V$ over $S$ which identifies
    $X_{V'}$ with the inverse image of $V'$ in $X_V$.
    We omit the verification that given affine opens
    $V'' \subset V' \subset V$ of $S$ we have
    $\rho^V_{V''} = \rho^V_{V'} \circ \rho^{V'}_{V''}$.
    
    \medskip\noindent
    By Constructions, Lemma \ref{constructions-lemma-relative-glueing} the data
    $(X_V, \rho^V_{V'})$ glue to a scheme $X \to S$.
    Moreover, we are given isomorphisms $V \times_S X \to X_V$
    which recover the maps $\rho^V_{V'}$. Unwinding the construction
    of the schemes $X_V$ we obtain isomorphisms
    $$
    V \times_S U_i \times_S X
    \longrightarrow
    V \times_S X_i
    $$
    compatible with the maps $\varphi_{ii'}$ and compatible with
    restricting to smaller affine opens in $X$. This implies that
    the canonical descent datum on $U_i \times_S X$ is isomorphic
    to the given descent datum and we win.
    \end{proof}

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