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Tag 0APJ

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

Lemma 39.15.2. Suppose given a cartesian diagram $$ \xymatrix{ X \ar[d]_f \ar[r] & \mathop{\rm Spec}(B) \ar[d] \\ Y \ar[r] & \mathop{\rm Spec}(A) } $$ of schemes. Let $E \subset Y$ be an intersection of a nonempty family of quasi-compact opens of $Y$. Then $$ \Gamma(f^{-1}(E), \mathcal{O}_X|_{f^{-1}(E)}) = \Gamma(E, \mathcal{O}_Y|_E) \otimes_A B $$ provided $Y$ is quasi-separated and $A \to B$ is flat.

Proof. Write $E = \bigcap_{i \in I} V_i$ with $V_i \subset Y$ quasi-compact open. We may and do assume that for $i, j \in I$ there exists a $k \in I$ with $V_k \subset V_i \cap V_j$. Then we have similarly that $f^{-1}(E) = \bigcap_{i \in I} f^{-1}(V_i)$ in $X$. Thus the result follows from equation (39.15.0.1) and the corresponding result for $V_i$ and $f^{-1}(V_i)$ which is Cohomology of Schemes, Lemma 29.5.2. $\square$

    The code snippet corresponding to this tag is a part of the file more-groupoids.tex and is located in lines 3063–3079 (see updates for more information).

    \begin{lemma}
    \label{lemma-affine-base-change}
    Suppose given a cartesian diagram
    $$
    \xymatrix{
    X \ar[d]_f \ar[r] & \Spec(B) \ar[d] \\
    Y \ar[r] & \Spec(A)
    }
    $$
    of schemes. Let $E \subset Y$ be an intersection of a nonempty family
    of quasi-compact opens of $Y$. Then
    $$
    \Gamma(f^{-1}(E), \mathcal{O}_X|_{f^{-1}(E)}) =
    \Gamma(E, \mathcal{O}_Y|_E) \otimes_A B
    $$
    provided $Y$ is quasi-separated and $A \to B$ is flat.
    \end{lemma}
    
    \begin{proof}
    Write $E = \bigcap_{i \in I} V_i$ with $V_i \subset Y$ quasi-compact open.
    We may and do assume that for $i, j \in I$ there exists a $k \in I$ with
    $V_k \subset V_i \cap V_j$. Then we have similarly that
    $f^{-1}(E) = \bigcap_{i \in I} f^{-1}(V_i)$ in $X$.
    Thus the result follows from equation (\ref{equation-sections-of-intersection})
    and the corresponding result for $V_i$ and $f^{-1}(V_i)$ which is
    Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}.
    \end{proof}

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