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

Lemma 39.15.1. Let $X$ be an ind-quasi-affine scheme. Let $E \subset X$ be an intersection of a nonempty family of quasi-compact opens of $X$. Set $A = \Gamma(E, \mathcal{O}_X|_E)$ and $Y = \mathop{\rm Spec}(A)$. Then the canonical morphism $$ j : (E, \mathcal{O}_X|_E) \longrightarrow (Y, \mathcal{O}_Y) $$ of Schemes, Lemma 25.6.4 determines an isomorphism $(E, \mathcal{O}_X|_E) \to (E', \mathcal{O}_Y|_{E'})$ where $E' \subset Y$ is an intersection of quasi-compact opens. If $W \subset E$ is open in $X$, then $j(W)$ is open in $Y$.

Proof. Note that $(E, \mathcal{O}_X|_E)$ is a locally ringed space so that Schemes, Lemma 25.6.4 applies to $A \to \Gamma(E, \mathcal{O}_X|_E)$. Write $E = \bigcap_{i \in I} U_i$ with $I \not = \emptyset$ and $U_i \subset X$ quasi-compact open. We may and do assume that for $i, j \in I$ there exists a $k \in I$ with $U_k \subset U_i \cap U_j$. Set $A_i = \Gamma(U_i, \mathcal{O}_{U_i})$. We obtain commutative diagrams $$ \xymatrix{ (E, \mathcal{O}_X|_E) \ar[r] \ar[d] & (\mathop{\rm Spec}(A), \mathcal{O}_{\mathop{\rm Spec}(A)}) \ar[d] \\ (U_i, \mathcal{O}_{U_i}) \ar[r] & (\mathop{\rm Spec}(A_i), \mathcal{O}_{\mathop{\rm Spec}(A_i)}) } $$ Since $U_i$ is quasi-affine, we see that $U_i \to \mathop{\rm Spec}(A_i)$ is a quasi-compact open immersion. On the other hand $A = \mathop{\rm colim}\nolimits A_i$. Hence $\mathop{\rm Spec}(A) = \mathop{\rm lim}\nolimits \mathop{\rm Spec}(A_i)$ as topological spaces (Limits, Lemma 31.4.6). Since $E = \mathop{\rm lim}\nolimits U_i$ (by Topology, Lemma 5.24.7) we see that $E \to \mathop{\rm Spec}(A)$ is a homeomorphism onto its image $E'$ and that $E'$ is the intersection of the inverse images of the opens $U_i \subset \mathop{\rm Spec}(A_i)$ in $\mathop{\rm Spec}(A)$. For any $e \in E$ the local ring $\mathcal{O}_{X, e}$ is the value of $\mathcal{O}_{U_i, e}$ which is the same as the value on $\mathop{\rm Spec}(A)$.

To prove the final assertion of the lemma we argue as follows. Pick $i, j \in I$ with $U_i \subset U_j$. Consider the following commutative diagrams $$ \xymatrix{ U_i \ar[r] \ar[d] & \mathop{\rm Spec}(A_i) \ar[d] \\ U_i \ar[r] & \mathop{\rm Spec}(A_j) } \quad\quad \xymatrix{ W \ar[r] \ar[d] & \mathop{\rm Spec}(A_i) \ar[d] \\ W \ar[r] & \mathop{\rm Spec}(A_j) } \quad\quad \xymatrix{ W \ar[r] \ar[d] & \mathop{\rm Spec}(A) \ar[d] \\ W \ar[r] & \mathop{\rm Spec}(A_j) } $$ By Properties, Lemma 27.18.4 the first diagram is cartesian. Hence the second is cartesian as well. Passing to the limit we find that the third diagram is cartesian, so the top horizontal arrow of this diagram is an open immersion. $\square$

    The code snippet corresponding to this tag is a part of the file more-groupoids.tex and is located in lines 2993–3007 (see updates for more information).

    \begin{lemma}
    \label{lemma-sits-in-functions}
    Let $X$ be an ind-quasi-affine scheme. Let $E \subset X$ be an
    intersection of a nonempty family of quasi-compact opens of $X$.
    Set $A = \Gamma(E, \mathcal{O}_X|_E)$ and $Y = \Spec(A)$.
    Then the canonical morphism
    $$
    j : (E, \mathcal{O}_X|_E) \longrightarrow (Y, \mathcal{O}_Y)
    $$
    of Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}
    determines an isomorphism
    $(E, \mathcal{O}_X|_E) \to (E', \mathcal{O}_Y|_{E'})$
    where $E' \subset Y$ is an intersection of quasi-compact opens.
    If $W \subset E$ is open in $X$, then $j(W)$ is open in $Y$.
    \end{lemma}
    
    \begin{proof}
    Note that $(E, \mathcal{O}_X|_E)$ is a locally ringed space so that
    Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} applies
    to $A \to \Gamma(E, \mathcal{O}_X|_E)$. Write $E = \bigcap_{i \in I} U_i$
    with $I \not = \emptyset$ and $U_i \subset X$ quasi-compact open.
    We may and do assume that for $i, j \in I$ there exists a $k \in I$ with
    $U_k \subset U_i \cap U_j$. Set $A_i = \Gamma(U_i, \mathcal{O}_{U_i})$.
    We obtain commutative diagrams
    $$
    \xymatrix{
    (E, \mathcal{O}_X|_E) \ar[r] \ar[d] &
    (\Spec(A), \mathcal{O}_{\Spec(A)}) \ar[d] \\
    (U_i, \mathcal{O}_{U_i}) \ar[r] &
    (\Spec(A_i), \mathcal{O}_{\Spec(A_i)})
    }
    $$
    Since $U_i$ is quasi-affine, we see that $U_i \to \Spec(A_i)$
    is a quasi-compact open immersion. On the other hand
    $A = \colim A_i$. Hence $\Spec(A) = \lim \Spec(A_i)$ as topological
    spaces (Limits, Lemma \ref{limits-lemma-topology-limit}). Since
    $E = \lim U_i$ (by Topology, Lemma \ref{topology-lemma-make-spectral-space})
    we see that $E \to \Spec(A)$ is a homeomorphism onto its
    image $E'$ and that $E'$ is the intersection of the inverse images
    of the opens $U_i \subset \Spec(A_i)$ in $\Spec(A)$. For any
    $e \in E$ the local ring $\mathcal{O}_{X, e}$ is the value
    of $\mathcal{O}_{U_i, e}$ which is the same as the value on $\Spec(A)$.
    
    \medskip\noindent
    To prove the final assertion of the lemma we argue as follows.
    Pick $i, j \in I$ with $U_i \subset U_j$.
    Consider the following commutative diagrams
    $$
    \xymatrix{
    U_i \ar[r] \ar[d] & \Spec(A_i) \ar[d] \\
    U_i \ar[r] & \Spec(A_j)
    }
    \quad\quad
    \xymatrix{
    W \ar[r] \ar[d] & \Spec(A_i) \ar[d] \\
    W \ar[r] & \Spec(A_j)
    }
    \quad\quad
    \xymatrix{
    W \ar[r] \ar[d] & \Spec(A) \ar[d] \\
    W \ar[r] & \Spec(A_j)
    }
    $$
    By Properties, Lemma
    \ref{properties-lemma-cartesian-diagram-quasi-affine}
    the first diagram is cartesian. Hence the second is cartesian as well.
    Passing to the limit we find that the third diagram
    is cartesian, so the top horizontal arrow of this diagram is an open immersion.
    \end{proof}

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