# The ring of global sections of a regular scheme

Let $$X$$ be a Noetherian regular scheme. Is $$\mathcal{O}_X(X)$$ a regular ring? For affine schemes this is true, see 02IU on the Stacks project.

The answer is no. For instance, take the quadratic cone $$Y = \mathrm{Spec}(\Bbbk[x,y,z]/(xz-y^2))$$ and let $$X$$ be its blowup at the vertex. Then $$X$$ is regular, but $$H^0(X,\mathcal{O}_X) = H^0(Y,\mathcal{O}_Y) = \Bbbk[x,y,z]/(xz-y^2)$$ is not.