Lemma 37.15.2 (047C)—The Stacks project

Lemma 37.15.2. Let $S$ be a scheme. Let

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } \]

be a cartesian diagram of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $x' \in X'$ with images $x = g'(x')$ and $s' = f'(x')$.

If $\mathcal{F}$ is flat over $S$ at $x$, then $(g')^*\mathcal{F}$ is flat over $S'$ at $x'$.
If $g$ is flat at $s'$ and $(g')^*\mathcal{F}$ is flat over $S'$ at $x'$, then $\mathcal{F}$ is flat over $S$ at $x$.

In particular, if $g$ is flat, $f$ is locally of finite presentation, and $\mathcal{F}$ is locally of finite presentation, then formation of the open subset of Theorem 37.15.1 commutes with base change.

Proof. Consider the commutative diagram of local rings

\[ \xymatrix{ \mathcal{O}_{X', x'} & \mathcal{O}_{X, x} \ar[l] \\ \mathcal{O}_{S', s'} \ar[u] & \mathcal{O}_{S, s} \ar[l] \ar[u] } \]

Note that $\mathcal{O}_{X', x'}$ is a localization of $\mathcal{O}_{X, x} \otimes _{\mathcal{O}_{S, s}} \mathcal{O}_{S', s'}$, and that $((g')^*\mathcal{F})_{x'}$ is equal to $\mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \mathcal{O}_{X', x'}$. Hence the lemma follows from Algebra, Lemma 10.100.1. $\square$

Comments (2)

Comment #7756 by James on September 13, 2022 at 05:54

In the statement we put $s' = f'(x')$ instead of $g'$ .

Comment #8002 by Stacks Project on January 16, 2023 at 11:28

Already fixed.

Comments (2)

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