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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').

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

  2. 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

In the statement we put instead of .


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