# The Stacks Project

## Tag 0APJ

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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