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The Stacks project

Lemma 38.10.5. Let f : X \to S be a morphism of schemes. Let \mathcal{F} be a quasi-coherent sheaf on X. Let x \in X with image s \in S. Assume that

  1. f is locally of finite type,

  2. \mathcal{F} is of finite type, and

  3. \mathcal{F} is flat at x over S.

Then there exists an elementary étale neighbourhood (S', s') \to (S, s) and an open subscheme

V \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})

which contains the unique point of X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'}) mapping to x such that the pullback of \mathcal{F} to V is flat over \mathcal{O}_{S', s'}.

Proof. (The only difference between this and Proposition 38.10.3 is that we do not assume f is of finite presentation.) The question is local on X and S, hence we may assume X and S are affine. Write X = \mathop{\mathrm{Spec}}(B), S = \mathop{\mathrm{Spec}}(A) and write B = A[x_1, \ldots , x_ n]/I. In other words we obtain a closed immersion i : X \to \mathbf{A}^ n_ S. Denote t = i(x) \in \mathbf{A}^ n_ S. We may apply Proposition 38.10.3 to \mathbf{A}^ n_ S \to S, the sheaf i_*\mathcal{F} and the point t. We obtain an elementary étale neighbourhood (S', s') \to (S, s) and an open subscheme

W \subset \mathbf{A}^ n_{\mathcal{O}_{S', s'}}

such that the pullback of i_*\mathcal{F} to W is flat over \mathcal{O}_{S', s'}. This means that V := W \cap \big (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})\big ) is the desired open subscheme. \square


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