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Chapter 39: More on Groupoid Schemes > Section 39.15: Descending ind-quasi-affine morphisms

Lemma 39.15.3 (Gabber). Let $S$ be a scheme. Let $\{X_i \to S\}_{i\in I}$ be an fpqc covering. Let $(V_i/X_i, \varphi_{ij})$ be a descent datum relative to $\{X_i \to S\}$, see Descent, Definition 34.31.3. If each morphism $V_i \to X_i$ is ind-quasi-affine, then the descent datum is effective.

Proof. Being ind-quasi-affine is a property of morphisms of schemes which is preserved under any base change, see More on Morphisms, Lemma 36.55.2. Hence Descent, Lemma 34.33.2 applies and it suffices to prove the statement of the lemma in case the fpqc-covering is given by a single $\{X \to S\}$ flat surjective morphism of affines. Say $X = \mathop{\rm Spec}(A)$ and $S = \mathop{\rm Spec}(R)$ so that $R \to A$ is a faithfully flat ring map. Let $(V, \varphi)$ be a descent datum relative to $X$ over $S$ and assume that $V \to X$ is ind-quasi-affine, in other words, $V$ is ind-quasi-affine.

Let $(U, R, s, t, c)$ be the groupoid scheme over $S$ with $U = X$ and $R = X \times_S X$ and $s$, $t$, $c$ as usual. By Groupoids, Lemma 38.21.3 the pair $(V, \varphi)$ corresponds to a cartesian morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoid schemes. Let $u' \in U'$ be any point. By Groupoids, Lemmas 38.19.2, 38.19.3, and 38.19.4 we can choose $u' \in W \subset E \subset U'$ where $W$ is open and $R'$-invariant, and $E$ is set-theoretically $R'$-invariant and an intersection of a nonempty family of quasi-compact opens.

Translating back to $(V, \varphi)$, for any $v \in V$ we can find $v \in W \subset E \subset V$ with the following properties: (a) $W$ is open and $\varphi(W \times_S X) = X \times_S W$ and (b) $E$ an intersection of quasi-compact opens and $\varphi(E \times_S X) = X \times_S E$ set-theoretically. Here we use the notation $E \times_S X$ to mean the inverse image of $E$ in $V \times_S X$ by the projection morphism and similarly for $X \times_S E$. By Lemma 39.15.2 this implies that $\varphi$ defines an isomorphism \begin{align*} \Gamma(E, \mathcal{O}_V|_E) \otimes_R A & = \Gamma(E \times_S X, \mathcal{O}_{V \times_S X}|_{E \times_S X}) \\ & \to \Gamma(X \times_S E, \mathcal{O}_{X \times_S V}|_{X \times_S E}) \\ & = A \otimes_R \Gamma(E, \mathcal{O}_V|_E) \end{align*} of $A \otimes_R A$-algebras which we will call $\psi$. The cocycle condition for $\varphi$ translates into the cocycle condition for $\psi$ as in Descent, Definition 34.3.1 (details omitted). By Descent, Proposition 34.3.9 we find an $R$-algebra $R'$ and an isomorphism $\chi : R' \otimes_R A \to \Gamma(E, \mathcal{O}_V|_E)$ of $A$-algebras, compatible with $\psi$ and the canonical descent datum on $R' \otimes_R A$.

By Lemma 39.15.1 we obtain a canonical ''embedding'' $$ j : (E, \mathcal{O}_V|_E) \longrightarrow \mathop{\rm Spec}(\Gamma(E, \mathcal{O}_V|_E)) = \mathop{\rm Spec}(R' \otimes_R A) $$ of locally ringed spaces. The construction of this map is canonical and we get a commutative diagram $$ \xymatrix{ & E \times_S X \ar[rr]_\varphi \ar[ld] \ar[rd]^{j'} & & X \times_S E \ar[rd] \ar[ld]_{j''} \\ E \ar[rd]^j & & \mathop{\rm Spec}(R' \otimes_R A \otimes_R A) \ar[ld] \ar[rd] & & E \ar[ld]_j \\ & \mathop{\rm Spec}(R' \otimes_R A) \ar[rd] && \mathop{\rm Spec}(R' \otimes_R A) \ar[ld] \\ & & \mathop{\rm Spec}(R') } $$ where $j'$ and $j''$ come from the same construction applied to $E \times_S X \subset V \times_S X$ and $X \times_S E \subset X \times_S V$ via $\chi$ and the identifications used to construct $\psi$. It follows that $j(W)$ is an open subscheme of $\mathop{\rm Spec}(R' \otimes_R A)$ whose inverse image under the two projections $\mathop{\rm Spec}(R' \otimes_R A \otimes_R A) \to \mathop{\rm Spec}(R' \otimes_R A)$ are equal. By Descent, Lemma 34.10.2 we find an open $W_0 \subset \mathop{\rm Spec}(R')$ whose base change to $\mathop{\rm Spec}(A)$ is $j(W)$. Contemplating the diagram above we see that the descent datum $(W, \varphi|_{W \times_S X})$ is effective. By Descent, Lemma 34.32.13 we see that our descent datum is effective. $\square$

    The code snippet corresponding to this tag is a part of the file more-groupoids.tex and is located in lines 3091–3099 (see updates for more information).

    \begin{lemma}[Gabber]
    \label{lemma-ind-quasi-affine}
    Let $S$ be a scheme. Let $\{X_i \to S\}_{i\in I}$ be an fpqc covering.
    Let $(V_i/X_i, \varphi_{ij})$ be a descent datum relative to
    $\{X_i \to S\}$, see Descent, Definition
    \ref{descent-definition-descent-datum-for-family-of-morphisms}. 
    If each morphism $V_i \to X_i$ is ind-quasi-affine, then the descent datum
    is effective.
    \end{lemma}
    
    \begin{proof}
    Being ind-quasi-affine is a property of morphisms of schemes
    which is preserved under any base change, see
    More on Morphisms, Lemma
    \ref{more-morphisms-lemma-base-change-ind-quasi-affine}.
    Hence Descent, Lemma \ref{descent-lemma-descending-types-morphisms} applies
    and it suffices to prove the statement of the lemma
    in case the fpqc-covering is given by a single
    $\{X \to S\}$ flat surjective morphism of affines.
    Say $X = \Spec(A)$ and $S = \Spec(R)$ so
    that $R \to A$ is a faithfully flat ring map.
    Let $(V, \varphi)$ be a descent datum relative to $X$ over $S$
    and assume that $V \to X$ is ind-quasi-affine, in other words,
    $V$ is ind-quasi-affine.
    
    \medskip\noindent
    Let $(U, R, s, t, c)$ be the groupoid scheme over $S$ with
    $U = X$ and $R = X \times_S X$ and $s$, $t$, $c$ as usual.
    By Groupoids, Lemma \ref{groupoids-lemma-cartesian-equivalent-descent-datum}
    the pair $(V, \varphi)$ corresponds to a cartesian morphism
    $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoid schemes.
    Let $u' \in U'$ be any point. By
    Groupoids, Lemmas \ref{groupoids-lemma-constructing-invariant-opens},
    \ref{groupoids-lemma-first-observation}, and
    \ref{groupoids-lemma-second-observation}
    we can choose $u' \in W \subset E \subset U'$
    where $W$ is open and $R'$-invariant, and
    $E$ is set-theoretically $R'$-invariant and
    an intersection of a nonempty family of quasi-compact opens.
    
    \medskip\noindent
    Translating back to $(V, \varphi)$, for any $v \in V$ we can find
    $v \in W \subset E \subset V$ with the following properties:
    (a) $W$ is open and $\varphi(W \times_S X) = X \times_S W$ and
    (b) $E$ an intersection of quasi-compact opens and
    $\varphi(E \times_S X) = X \times_S E$ set-theoretically.
    Here we use the notation $E \times_S X$ to mean the
    inverse image of $E$ in $V \times_S X$ by the projection morphism and
    similarly for $X \times_S E$. By Lemma \ref{lemma-affine-base-change}
    this implies that $\varphi$ defines an isomorphism
    \begin{align*}
    \Gamma(E, \mathcal{O}_V|_E) \otimes_R A
    & =
    \Gamma(E \times_S X, \mathcal{O}_{V \times_S X}|_{E \times_S X}) \\
    & \to
    \Gamma(X \times_S E, \mathcal{O}_{X \times_S V}|_{X \times_S E}) \\
    & =
    A \otimes_R \Gamma(E, \mathcal{O}_V|_E)
    \end{align*}
    of $A \otimes_R A$-algebras which we will call $\psi$.
    The cocycle condition for $\varphi$
    translates into the cocycle condition for $\psi$ as in
    Descent, Definition \ref{descent-definition-descent-datum-modules}
    (details omitted). By Descent, Proposition
    \ref{descent-proposition-descent-module}
    we find an $R$-algebra $R'$ and an isomorphism
    $\chi : R' \otimes_R A \to \Gamma(E, \mathcal{O}_V|_E)$
    of $A$-algebras, compatible with $\psi$ and the
    canonical descent datum on $R' \otimes_R A$.
    
    \medskip\noindent
    By Lemma \ref{lemma-sits-in-functions} we obtain a canonical ``embedding''
    $$
    j : (E, \mathcal{O}_V|_E) \longrightarrow
    \Spec(\Gamma(E, \mathcal{O}_V|_E)) = \Spec(R' \otimes_R A)
    $$
    of locally ringed spaces. The construction of this map is canonical
    and we get a commutative diagram
    $$
    \xymatrix{
    & E \times_S X \ar[rr]_\varphi \ar[ld] \ar[rd]^{j'} & &
    X \times_S E \ar[rd] \ar[ld]_{j''} \\
    E \ar[rd]^j  & &
    \Spec(R' \otimes_R A \otimes_R A) \ar[ld] \ar[rd] & &
    E \ar[ld]_j \\
    & \Spec(R' \otimes_R A) \ar[rd] && \Spec(R' \otimes_R A) \ar[ld] \\
    & & \Spec(R')
    }
    $$
    where $j'$ and $j''$ come from the same construction applied to
    $E \times_S X \subset V \times_S X$ and $X \times_S E \subset X \times_S V$
    via $\chi$ and the identifications used to construct $\psi$.
    It follows that $j(W)$ is an open subscheme of $\Spec(R' \otimes_R A)$
    whose inverse image under the two projections
    $\Spec(R' \otimes_R A \otimes_R A) \to \Spec(R' \otimes_R A)$
    are equal. By Descent, Lemma \ref{descent-lemma-open-fpqc-covering}
    we find an open $W_0 \subset \Spec(R')$ whose base change
    to $\Spec(A)$ is $j(W)$. Contemplating the diagram above
    we see that the descent datum $(W, \varphi|_{W \times_S X})$
    is effective. By Descent, Lemma
    \ref{descent-lemma-effective-for-fpqc-is-local-upstairs}
    we see that our descent datum is effective.
    \end{proof}

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