I have showed that every open/closed immersion of schemes is a monomorphism of schemes, but I know that the converse is not true, i.e., not every monomorphism of schemes is an open/closed immersion. However, I have not been able to find an counterexample. Does anyone know a monomorphism of schemes that is not a composition of open or closed immersions?
Thanks in advance.
On
Example 1: Let $X$ be the curve obtained by gluing the points $0$ with $\infty$ in $\mathbb{P}^1$. We have a natural normalization morphism $f: \mathbb{P}^1\to X$. The restriction of $f$ to $\mathbb{A}^1$, which we call $g: \mathbb{A}^1\to X$ is a monomorphism of schemes:
Indeed, for any point $x\in X,$ we have that the fiber $g_x: \mathbb{A}^1\times_X x\to x$ is an isomorphism. Thus, by 05VH, we have that $g$ is a monomorphism.
In this example, $g$ is an open immersion followed by a normalization morphism, not a closed immersion.
Example 2: Consider $0\coprod \mathbb{G}_m\to \mathbb{A}^1.$ By the same reasoning as above, we see that this morphism is also a monomorphism. It is a union of closed and open immersions, but not a composition of them.
As a composition of two monomorphisms is a monomorphism, you can take the composition of an open and a closed immersion. For example $$\mathbb A^1 \setminus \{0\} \hookrightarrow \mathbb A^2, x \mapsto (x,0).$$