Let $f:X\to Y$ be a smooth morphism of relative dimension $n$ of separated schemes which are of finite type over $\text{Spec}(k)$, where $k$ is any field. Suppose that $i: Y\to X$ is a section of $f$, i.e. $f\circ i = \text{id}_Y$. I want to prove that in this case, $i$ is a regular closed embedding of codimension $n$. By this I mean that $i$ is a closed embedding, and every point $y\in Y$ has an affine open neighborhood $\text{Spec}(A)$ in $X$ such that if $Y$ corresponds to an ideal $I\subset A$, we have that $I$ can be generated by a regular sequence of length $n$.
This statement is in Fultons 'Intersection Theory', Appendix B.7.3. It is also in the Stacks Project, Tag 067R. However, I do not completely understand the given proofs. So far I can prove the following:
- $i$ is a closed immersion.
- $i$ is regular if and only if for every closed point of $Y$, say corresponding to a maximal ideal $\mathfrak{m}$ in an affine open $\text{Spec}(A)$, the image $I_{\mathfrak{m}}$ of $I$ in the local ring $A_{\mathfrak{m}}$ can be generated by a regular sequence.
- For every closed point as above, the inclusion $\text{Spec}(A/\mathfrak{m})\to X$ is a regular embedding. So the embedding is regular on the fibers.
But how to conclude from this that $i$ is a regular embedding? Any help would be really appreciated.
You can deduce your statement from the previous appendix B.7.2 ( if you have proven this) which says that $i : X \rightarrow Y$ is closed imbedding and $ g : Y \rightarrow S$ any morphism such that $X $ is smooth over $S$. Then $i$ is regular imbedding iff $Y$ is smooth over $S$ in some neighbourhood of $X$. In your case, composition is identity which is clearly smooth and $f$ is also smooth and you have proven the section is closed imbedding.