$\mathbb{P}^n_k$ is irreducible as a scheme

Question

How do I show that $\mathbb{P}^n_k$ is irreducible as a scheme? I am still quite confused with the definition by gluing affine schemes. I tried to show that nonempty open sets intersect, but got lost.

Answer 1

Actually, it’s not a scheme question, it really is a topological one.

There are open subsets $U_0,\ldots,U_n$, all homeomorphic and irreducible (they’re the $D(X_i) \cong \mathbb{A}^n_k$) with nonempty global intersection $V$ such that they cover $\mathbb{P}^n_k$.

Let $A,B \subset \mathbb{P}^n_k$ be nonempty open subsets.

Say, $A’=A \cap U_i \neq \emptyset$. Then $A’$ is a nonempty open subset of $U_i$ which is irreducible; moreover, $V \subset U_i$ is a nonempty open subset so $V \cap A’=V \cap A$ is nonempty.

Let $j$ be such that $B’’=B \cap U_j$ is nonempty. Let $A’’ = A \cap U_j \supset A \cap V$. Then $B’’, A’’$ are both nonempty open subsets of $U_j$ so they must meet; as $A \cap B \supset A’’ \cap B’’$, $A$ and $B$ meet and we are done.