Let $X$ be the underlying space of a scheme.
- If $X$ is irreducible of finite Krull dimension, is it necessarily quasi-compact?
- Is it necessarily Noetherian?
- What if we assume not only that Krull dimension is finite but also that it is 1?
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
Sign up to join this communityLet $X$ be the underlying space of a scheme.
The answer is no for all these questions. Take the line with infinite origins: the scheme obtained by gluing an infinite amount of copies of $\mathbb{A}^1$ along the open subsets $\mathbb{G}_m$. This has Krull dimension 1 (there are only closed points and the unique generic point) and it is irreducible (the only proper nonempty closed subsets are the closed points) but it is not quasi-compact (it contains an infinite discrete set), and so in particular not Noetherian.