The Stacks project

101.44 Local complete intersection morphisms

The property “being a local complete intersection morphism” of morphisms of algebraic spaces is smooth local on the source-and-target, see Descent on Spaces, Lemma 74.20.4 and More on Morphisms of Spaces, Lemmas 76.48.9 and 76.48.10. By Lemma 101.16.1 above, we may define what it means for a morphism of algebraic spaces to be a local complete intersection morphism as follows and it agrees with the already existing notion defined in More on Morphisms of Spaces, Section 76.48 when both source and target are algebraic spaces.

Definition 101.44.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is a local complete intersection morphism or Koszul if the equivalent conditions of Lemma 101.16.1 hold with $\mathcal{P} = \text{local complete intersection}$.

Lemma 101.44.2. The composition of local complete intersection morphisms is a local complete intersection.

Proof. Combine Remark 101.16.3 with More on Morphisms of Spaces, Lemma 76.48.5. $\square$

Lemma 101.44.3. A flat base change of a local complete intersection morphism is a local complete intersection morphism.

Proof. Omitted. Hint: Argue exactly as in Remark 101.16.4 (but only for flat $\mathcal{Y}' \to \mathcal{Y}$) using More on Morphisms of Spaces, Lemma 76.48.4. $\square$

Lemma 101.44.4. Let

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

be a commutative diagram of morphisms of algebraic stacks. Assume $\mathcal{Y} \to \mathcal{Z}$ is smooth and $\mathcal{X} \to \mathcal{Z}$ is a local complete intersection morphism. Then $f : \mathcal{X} \to \mathcal{Y}$ is a local complete intersection morphism.

Proof. Choose a scheme $W$ and a surjective smooth morphism $W \to \mathcal{Z}$. Choose a scheme $V$ and a surjective smooth morphism $V \to W \times _\mathcal {Z} \mathcal{Y}$. Choose a scheme $U$ and a surjective smooth morphism $U \to V \times _\mathcal {Y} \mathcal{X}$. Then $U \to W$ is a local complete intersection morphism of schemes and $V \to W$ is a smooth morphism of schemes. By the result for schemes (More on Morphisms, Lemma 37.62.10) we conclude that $U \to V$ is a local complete intersection morphism. By definition this means that $f$ is a local complete intersection morphism. $\square$


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Your email address will not be published.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 0CJ2. Beware of the difference between the letter 'O' and the digit '0'.