The Stacks project

Lemma 77.11.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $n \geq 0$. The following are equivalent

  1. for some commutative diagram

    \[ \xymatrix{ U \ar[d]_\varphi \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

    with surjective, ├ętale vertical arrows where $U$ and $V$ are schemes, the sheaf $\varphi ^*\mathcal{F}$ is flat over $V$ in dimensions $\geq n$ (More on Flatness, Definition 38.20.10),

  2. for every commutative diagram

    \[ \xymatrix{ U \ar[d]_\varphi \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

    with ├ętale vertical arrows where $U$ and $V$ are schemes, the sheaf $\varphi ^*\mathcal{F}$ is flat over $V$ in dimensions $\geq n$, and

  3. for $x \in |X|$ such that $\mathcal{F}$ is not flat at $x$ over $Y$ the transcendence degree of $x/f(x)$ is $< n$ (Morphisms of Spaces, Definition 67.33.1).

If this is true, then it remains true after any base change $Y' \to Y$.

Proof. Suppose that we have a diagram as in (1). Then the equivalence of the conditions in More on Flatness, Lemma 38.20.9 shows that (1) and (3) are equivalent. But condition (3) is inherited by $\varphi ^*\mathcal{F}$ for any $U \to V$ as in (2). Whence we see that (3) implies (2) by the result for schemes again. The result for schemes also implies the statement on base change. $\square$

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