# Krull dimension of the ring of global sections

Let $$X$$ be an irreducible scheme. Can the Krull dimension of $$\mathcal{O}_X(X)$$ exceed that of $$X$$?

• – Yemon Choi May 6 at 22:55

Yes, this happens if $$X$$ is the punctured spectrum of a two dimensional regular local ring.
• what is the Krull dimension of such $X$? I believe $X$ has closed points, so puncturing should not really change anything, right? – user138661 May 4 at 20:19
• @schematic_boi The Krull dimension of $X$ is $1$. – Mere Scribe May 4 at 23:50
• can this happen if $X$ is a scheme locally of finite type over $\mathbb{Z}$? $X$ of finite type over $\mathbb{Z}$? – user138661 May 5 at 5:49