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 ├ętale 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 ├ętale 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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