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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CWS. Beware of the difference between the letter 'O' and the digit '0'.