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Tag 023R

34.5. Fpqc descent of quasi-coherent sheaves

The main application of flat descent for modules is the corresponding descent statement for quasi-coherent sheaves with respect to fpqc-coverings.

Lemma 34.5.1. Let $S$ be an affine scheme. Let $\mathcal{U} = \{f_i : U_i \to S\}_{i = 1, \ldots, n}$ be a standard fpqc covering of $S$, see Topologies, Definition 33.9.9. Any descent datum on quasi-coherent sheaves for $\mathcal{U} = \{U_i \to S\}$ is effective. Moreover, the functor from the category of quasi-coherent $\mathcal{O}_S$-modules to the category of descent data with respect to $\mathcal{U}$ is fully faithful.

Proof. This is a restatement of Proposition 34.3.9 in terms of schemes. First, note that a descent datum $\xi$ for quasi-coherent sheaves with respect to $\mathcal{U}$ is exactly the same as a descent datum $\xi'$ for quasi-coherent sheaves with respect to the covering $\mathcal{U}' = \{\coprod_{i = 1, \ldots, n} U_i \to S\}$. Moreover, effectivity for $\xi$ is the same as effectivity for $\xi'$. Hence we may assume $n = 1$, i.e., $\mathcal{U} = \{U \to S\}$ where $U$ and $S$ are affine. In this case descent data correspond to descent data on modules with respect to the ring map $$ \Gamma(S, \mathcal{O}) \longrightarrow \Gamma(U, \mathcal{O}). $$ Since $U \to S$ is surjective and flat, we see that this ring map is faithfully flat. In other words, Proposition 34.3.9 applies and we win. $\square$

Proposition 34.5.2. Let $S$ be a scheme. Let $\mathcal{U} = \{\varphi_i : U_i \to S\}$ be an fpqc covering, see Topologies, Definition 33.9.1. Any descent datum on quasi-coherent sheaves for $\mathcal{U} = \{U_i \to S\}$ is effective. Moreover, the functor from the category of quasi-coherent $\mathcal{O}_S$-modules to the category of descent data with respect to $\mathcal{U}$ is fully faithful.

Proof. Let $S = \bigcup_{j \in J} V_j$ be an affine open covering. For $j, j' \in J$ we denote $V_{jj'} = V_j \cap V_{j'}$ the intersection (which need not be affine). For $V \subset S$ open we denote $\mathcal{U}_V = \{V \times_S U_i \to V\}_{i \in I}$ which is a fpqc-covering (Topologies, Lemma 33.9.7). By definition of an fpqc covering, we can find for each $j \in J$ a finite set $K_j$, a map $\underline{i} : K_j \to I$, affine opens $U_{\underline{i}(k), k} \subset U_{\underline{i}(k)}$, $k \in K_j$ such that $\mathcal{V}_j = \{U_{\underline{i}(k), k} \to V_j\}_{k \in K_j}$ is a standard fpqc covering of $V_j$. And of course, $\mathcal{V}_j$ is a refinement of $\mathcal{U}_{V_j}$. Picture $$ \xymatrix{ \mathcal{V}_j \ar[r] \ar@{~>}[d] & \mathcal{U}_{V_j} \ar[r] \ar@{~>}[d] & \mathcal{U} \ar@{~>}[d] \\ V_j \ar@{=}[r] & V_j \ar[r] & S } $$ where the top horizontal arrows are morphisms of families of morphisms with fixed target (see Sites, Definition 7.8.1).

To prove the proposition you show successively the faithfulness, fullness, and essential surjectivity of the functor from quasi-coherent sheaves to descent data.

Faithfulness. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent sheaves on $S$ and let $a, b : \mathcal{F} \to \mathcal{G}$ be homomorphisms of $\mathcal{O}_S$-modules. Suppose $\varphi_i^*(a) = \varphi^*(b)$ for all $i$. Pick $s \in S$. Then $s = \varphi_i(u)$ for some $i \in I$ and $u \in U_i$. Since $\mathcal{O}_{S, s} \to \mathcal{O}_{U_i, u}$ is flat, hence faithfully flat (Algebra, Lemma 10.38.17) we see that $a_s = b_s : \mathcal{F}_s \to \mathcal{G}_s$. Hence $a = b$.

Fully faithfulness. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent sheaves on $S$ and let $a_i : \varphi_i^*\mathcal{F} \to \varphi_i^*\mathcal{G}$ be homomorphisms of $\mathcal{O}_{U_i}$-modules such that $\text{pr}_0^*a_i = \text{pr}_1^*a_j$ on $U_i \times_U U_j$. We can pull back these morphisms to get morphisms $$ a_k : \mathcal{F}|_{U_{\underline{i}(k), k}} \longrightarrow \mathcal{G}|_{U_{\underline{i}(k), k}} $$ $k \in K_j$ with notation as above. Moreover, Lemma 34.2.2 assures us that these define a morphism between (canonical) descent data on $\mathcal{V}_j$. Hence, by Lemma 34.5.1, we get correspondingly unique morphisms $a_j : \mathcal{F}|_{V_j} \to \mathcal{G}|_{V_j}$. To see that $a_j|_{V_{jj'}} = a_{j'}|_{V_{jj'}}$ we use that both $a_j$ and $a_{j'}$ agree with the pullback of the morphism $(a_i)_{i \in I}$ of (canonical) descent data to any covering refining both $\mathcal{V}_{j, V_{jj'}}$ and $\mathcal{V}_{j', V_{jj'}}$, and using the faithfulness already shown. For example the covering $\mathcal{V}_{jj'} = \{V_k \times_S V_{k'} \to V_{jj'}\}_{k \in K_j, k' \in K_{j'}}$ will do.

Essential surjectivity. Let $\xi = (\mathcal{F}_i, \varphi_{ii'})$ be a descent datum for quasi-coherent sheaves relative to the covering $\mathcal{U}$. Pull back this descent datum to get descent data $\xi_j$ for quasi-coherent sheaves relative to the coverings $\mathcal{V}_j$ of $V_j$. By Lemma 34.5.1 once again there exist quasi-coherent sheaves $\mathcal{F}_j$ on $V_j$ whose associated canonical descent datum is isomorphic to $\xi_j$. By fully faithfulness (proved above) we see there are isomorphisms $$ \phi_{jj'} : \mathcal{F}_j|_{V_{jj'}} \longrightarrow \mathcal{F}_{j'}|_{V_{jj'}} $$ corresponding to the isomorphism of descent data between the pullback of $\xi_j$ and $\xi_{j'}$ to $\mathcal{V}_{jj'}$. To see that these maps $\phi_{jj'}$ satisfy the cocycle condition we use faithfulness (proved above) over the triple intersections $V_{jj'j''}$. Hence, by Lemma 34.2.4 we see that the sheaves $\mathcal{F}_j$ glue to a quasi-coherent sheaf $\mathcal{F}$ as desired. We still have to verify that the canonical descent datum relative to $\mathcal{U}$ associated to $\mathcal{F}$ is isomorphic to the descent datum we started out with. This verification is omitted. $\square$

    The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 1532–1691 (see updates for more information).

    \section{Fpqc descent of quasi-coherent sheaves}
    \label{section-fpqc-descent-quasi-coherent}
    
    \noindent
    The main application of flat descent for modules is
    the corresponding descent statement for quasi-coherent
    sheaves with respect to fpqc-coverings.
    
    \begin{lemma}
    \label{lemma-standard-fpqc-covering}
    Let $S$ be an affine scheme.
    Let $\mathcal{U} = \{f_i : U_i \to S\}_{i = 1, \ldots, n}$
    be a standard fpqc covering of $S$, see
    Topologies, Definition \ref{topologies-definition-standard-fpqc}.
    Any descent datum on quasi-coherent sheaves
    for $\mathcal{U} = \{U_i \to S\}$ is effective.
    Moreover, the functor from the category of
    quasi-coherent $\mathcal{O}_S$-modules to the category
    of descent data with respect to $\mathcal{U}$ is fully faithful.
    \end{lemma}
    
    \begin{proof}
    This is a restatement of Proposition \ref{proposition-descent-module}
    in terms of schemes. First, note that a descent datum $\xi$
    for quasi-coherent sheaves with respect to $\mathcal{U}$
    is exactly the same as a descent datum $\xi'$ for quasi-coherent sheaves
    with respect to the covering
    $\mathcal{U}' = \{\coprod_{i = 1, \ldots, n} U_i \to S\}$.
    Moreover, effectivity for $\xi$ is the same as effectivity for $\xi'$.
    Hence we may assume $n = 1$, i.e., $\mathcal{U} = \{U \to S\}$
    where $U$ and $S$ are affine. In this case descent data
    correspond to descent data on modules with respect to the ring map
    $$
    \Gamma(S, \mathcal{O})
    \longrightarrow
    \Gamma(U, \mathcal{O}).
    $$
    Since $U \to S$ is surjective and flat, we see that this ring map
    is faithfully flat. In other words,
    Proposition \ref{proposition-descent-module} applies and we win.
    \end{proof}
    
    \begin{proposition}
    \label{proposition-fpqc-descent-quasi-coherent}
    Let $S$ be a scheme.
    Let $\mathcal{U} = \{\varphi_i : U_i \to S\}$ be an fpqc covering, see
    Topologies, Definition \ref{topologies-definition-fpqc-covering}.
    Any descent datum on quasi-coherent sheaves
    for $\mathcal{U} = \{U_i \to S\}$ is effective.
    Moreover, the functor from the category of
    quasi-coherent $\mathcal{O}_S$-modules to the category
    of descent data with respect to $\mathcal{U}$ is fully faithful.
    \end{proposition}
    
    \begin{proof}
    Let $S = \bigcup_{j \in J} V_j$ be an affine open covering.
    For $j, j' \in J$ we denote $V_{jj'} = V_j \cap V_{j'}$ the intersection
    (which need not be affine). For $V \subset S$ open we denote
    $\mathcal{U}_V = \{V \times_S U_i \to V\}_{i \in I}$ which is a
    fpqc-covering (Topologies, Lemma \ref{topologies-lemma-fpqc}).
    By definition of an fpqc covering, we can find for each $j \in J$ a
    finite set $K_j$, a map $\underline{i} : K_j \to I$,
    affine opens $U_{\underline{i}(k), k} \subset U_{\underline{i}(k)}$,
    $k \in K_j$ such that
    $\mathcal{V}_j = \{U_{\underline{i}(k), k} \to V_j\}_{k \in K_j}$ is
    a standard fpqc covering of $V_j$. And of course, $\mathcal{V}_j$
    is a refinement of $\mathcal{U}_{V_j}$. Picture
    $$
    \xymatrix{
    \mathcal{V}_j \ar[r] \ar@{~>}[d] &
    \mathcal{U}_{V_j} \ar[r] \ar@{~>}[d] &
    \mathcal{U} \ar@{~>}[d] \\
    V_j \ar@{=}[r] & V_j \ar[r] & S
    }
    $$
    where the top horizontal arrows are morphisms of families of
    morphisms with fixed target (see
    Sites, Definition \ref{sites-definition-morphism-coverings}).
    
    \medskip\noindent
    To prove the proposition you show successively the
    faithfulness, fullness, and essential surjectivity of the
    functor from quasi-coherent sheaves to descent data.
    
    \medskip\noindent
    Faithfulness. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent
    sheaves on $S$ and let $a, b : \mathcal{F} \to \mathcal{G}$ be
    homomorphisms of $\mathcal{O}_S$-modules.
    Suppose $\varphi_i^*(a) = \varphi^*(b)$ for all $i$.
    Pick $s \in S$. Then $s = \varphi_i(u)$ for some $i \in I$ and
    $u \in U_i$. Since $\mathcal{O}_{S, s} \to \mathcal{O}_{U_i, u}$
    is flat, hence faithfully flat
    (Algebra, Lemma \ref{algebra-lemma-local-flat-ff}) we see
    that $a_s = b_s : \mathcal{F}_s \to \mathcal{G}_s$. Hence $a = b$.
    
    \medskip\noindent
    Fully faithfulness. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent
    sheaves on $S$ and let
    $a_i : \varphi_i^*\mathcal{F} \to \varphi_i^*\mathcal{G}$ be
    homomorphisms of $\mathcal{O}_{U_i}$-modules such that
    $\text{pr}_0^*a_i = \text{pr}_1^*a_j$ on $U_i \times_U U_j$.
    We can pull back these morphisms to get morphisms
    $$
    a_k :
    \mathcal{F}|_{U_{\underline{i}(k), k}}
    \longrightarrow
    \mathcal{G}|_{U_{\underline{i}(k), k}}
    $$
    $k \in K_j$ with notation as above. Moreover,
    Lemma \ref{lemma-refine-descent-datum} assures us
    that these define a morphism between (canonical) descent data on
    $\mathcal{V}_j$. Hence, by
    Lemma \ref{lemma-standard-fpqc-covering}, we get correspondingly
    unique morphisms $a_j : \mathcal{F}|_{V_j} \to \mathcal{G}|_{V_j}$.
    To see that $a_j|_{V_{jj'}} = a_{j'}|_{V_{jj'}}$ we use that
    both $a_j$ and $a_{j'}$ agree with the pullback of the morphism
    $(a_i)_{i \in I}$ of (canonical) descent data to any covering
    refining both $\mathcal{V}_{j, V_{jj'}}$ and
    $\mathcal{V}_{j', V_{jj'}}$, and using the faithfulness already
    shown. For example the covering
    $\mathcal{V}_{jj'} =
    \{V_k \times_S V_{k'} \to V_{jj'}\}_{k \in K_j, k' \in K_{j'}}$
    will do.
    
    \medskip\noindent
    Essential surjectivity. Let $\xi = (\mathcal{F}_i, \varphi_{ii'})$
    be a descent datum for quasi-coherent sheaves relative to the covering
    $\mathcal{U}$. Pull back this descent datum to get descent data
    $\xi_j$ for quasi-coherent sheaves relative to the coverings
    $\mathcal{V}_j$ of $V_j$. By Lemma \ref{lemma-standard-fpqc-covering}
    once again there exist
    quasi-coherent sheaves $\mathcal{F}_j$ on $V_j$ whose associated
    canonical descent datum is isomorphic to $\xi_j$. By fully faithfulness
    (proved above) we see there are isomorphisms
    $$
    \phi_{jj'} :
    \mathcal{F}_j|_{V_{jj'}}
    \longrightarrow
    \mathcal{F}_{j'}|_{V_{jj'}}
    $$
    corresponding to the isomorphism of descent data between the pullback
    of $\xi_j$ and $\xi_{j'}$ to $\mathcal{V}_{jj'}$. To see that these
    maps $\phi_{jj'}$ satisfy the cocycle condition we use faithfulness
    (proved above) over the triple intersections $V_{jj'j''}$. Hence, by
    Lemma \ref{lemma-zariski-descent-effective}
    we see that the sheaves $\mathcal{F}_j$
    glue to a quasi-coherent sheaf $\mathcal{F}$ as desired.
    We still have to verify that the canonical descent datum relative to
    $\mathcal{U}$ associated to $\mathcal{F}$ is isomorphic to the descent
    datum we started out with. This verification is omitted.
    \end{proof}

    Comments (2)

    Comment #1534 by jojo on June 22, 2015 a 5:31 pm UTC

    I get a parsing error for the first equation of the proof.

    Comment #1536 by Pieter Belmans (site) on June 23, 2015 a 9:47 am UTC

    Thanks for noticing! It is now fixed.

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