The Stacks project

Lemma 70.6.12. Notation and assumptions as in Situation 70.6.1. Let $\mathcal{F}_0$ be a quasi-coherent $\mathcal{O}_{X_0}$-module and denote $\mathcal{F}_ i$ the pullback to $X_ i$ and $\mathcal{F}$ the pullback to $X$. If

  1. $\mathcal{F}$ is flat over $Y$,

  2. $\mathcal{F}_0$ is of finite presentation, and

  3. $f_0$ is locally of finite presentation,

then $\mathcal{F}_ i$ is flat over $Y_ i$ for some $i \geq 0$. In particular, if $f_0$ is locally of finite presentation and $f$ is flat, then $f_ i$ is flat for some $i \geq 0$.

Proof. Choose an affine scheme $V_0$ and a surjective étale morphism $V_0 \to Y_0$. Choose an affine scheme $U_0$ and a surjective étale morphism $U_0 \to V_0 \times _{Y_0} X_0$. Diagram

\[ \xymatrix{ U_0 \ar[d] \ar[r] & V_0 \ar[d] \\ X_0 \ar[r] & Y_0 } \]

The vertical arrows are surjective and étale by construction. We can base change this diagram to $B_ i$ or $B$ to get

\[ \vcenter { \xymatrix{ U_ i \ar[d] \ar[r] & V_ i \ar[d] \\ X_ i \ar[r] & Y_ i } } \quad \text{and}\quad \vcenter { \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } } \]

Note that $U_ i, V_ i, U, V$ are affine schemes, the vertical morphisms are surjective étale, and the limit of the morphisms $U_ i \to V_ i$ is $U \to V$. Recall that $\mathcal{F}_ i$ is flat over $Y_ i$ if and only if $\mathcal{F}_ i|_{U_ i}$ is flat over $V_ i$ and similarly $\mathcal{F}$ is flat over $Y$ if and only if $\mathcal{F}|_ U$ is flat over $V$ (Morphisms of Spaces, Definition 67.30.1). Since $f_0$ is locally of finite presentation, so is the morphism $U_0 \to V_0$. Hence the lemma follows from Limits, Lemma 32.10.4. $\square$

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