Chracterisation of Projective Line $\mathbb{P}^1_k$

Question

I have a question about an argument used in Prop 50.10.4 for chracterisation of projective line $\mathbb{P}^1_k$ from https://stacks.math.columbia.edu/tag/0C6L

Let $k$ be a field and $X$ be proper.

Consider the implication (6) -> (1)

According to the source this is a consequence from 50.10.2: https://stacks.math.columbia.edu/tag/0C6T

The problem is that 50.10.2 provides only a closed immersion $X \to \mathbb{P}^1_k$.

Why is it an isomorphism of schemes?

My considerations: Since both are irreducible curves of dimension $1$ we conclude that $X$ is homeomorphic to $\mathbb{P}^1_k$ (as topological space) and by 50.10.2 a closed immersion as scheme morphism.

But does this already imply that it is an iso of schemes?

Answer 1

A closed immersion of schemes $f:X\to \mathbb{P}^1_k$ means that on an affine open $U=Spec(R)\subset \mathbb{P}^1_k$, we have $f^{-1}(U)$ isomorphic to $Spec(R/I)$ for an ideal $I\subset R$. Since $X$ is one dimensional, and since closed subsets of $\mathbb{P}_k^1$ are either finite or the entire space, this should give what you want.