Lemma 40.15.2 (0APJ)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 40: More on Groupoid Schemes
Section 40.15: Descending ind-quasi-affine morphisms
Lemma 40.15.2 (cite)

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Lemma 40.15.2. Suppose given a cartesian diagram

\[ \xymatrix{ X \ar[d]_ f \ar[r] & \mathop{\mathrm{Spec}}(B) \ar[d] \\ Y \ar[r] & \mathop{\mathrm{Spec}}(A) } \]

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 (40.15.0.1) and the corresponding result for $V_ i$ and $f^{-1}(V_ i)$ which is Cohomology of Schemes, Lemma 30.5.2. $\square$

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