The Stacks project

Lemma 49.11.3. With notation as in Example 49.11.2 there is an open subscheme $U_ d \subset X_ d$ with the following property: a morphism of schemes $X \to X_ d$ factors through $U_ d$ if and only if $Y_ d \times _{X_ d} X \to X$ is syntomic.

Proof. Recall that being syntomic is the same thing as being flat and a local complete intersection morphism, see More on Morphisms, Lemma 37.62.8. The set $W_ d \subset Y_ d$ of points where $\pi _ d$ is Koszul is open in $Y_ d$ and its formation commutes with arbitrary base change, see More on Morphisms, Lemma 37.62.21. Since $\pi _ d$ is finite and hence closed, we see that $Z = \pi _ d(Y_ d \setminus W_ d)$ is closed. Since clearly $U_ d = X_ d \setminus Z$ and since its formation commutes with base change we find that the lemma is true. $\square$

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Your email address will not be published.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 0FL0. Beware of the difference between the letter 'O' and the digit '0'.