The Stacks project

Lemma 76.49.7. Consider a commutative diagram

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[ld]^ q \\ & Z } \]

of algebraic spaces. Assume that

  1. $p$ is flat and locally of finite presentation,

  2. $p$ is closed, and

  3. $q$ is flat and locally of finite presentation,

Then there exists an open subspace $W \subset Z$ such that a morphism $Z' \to Z$ factors through $W$ if and only if the base change $f_{Z'} : X_{Z'} \to Y_{Z'}$ is a local complete intersection morphism.

Proof. By Lemma 76.48.11 there exists an open subspace $U(f) \subset X$ which is the set of points where $f$ is Koszul. Moreover, formation of $U(f)$ commutes with arbitrary base change. Let $W \subset Z$ be the open subspace (see Properties of Spaces, Lemma 66.4.8) with underlying set of points

\[ |W| = |Z| \setminus |p|\left(|X| \setminus |U(f)|\right) \]

i.e., $z \in |Z|$ is a point of $W$ if and only if $f$ is Koszul at every point of $X$ above $z$. Note that this is open because we assumed that $p$ is closed. Since the formation of $U(f)$ commutes with arbitrary base change we immediately see (using Properties of Spaces, Lemma 66.4.9) that $W$ has the desired universal property. $\square$

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Your email address will not be published.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 06CE. Beware of the difference between the letter 'O' and the digit '0'.