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.
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6$\begingroup$ Probably related: meta.mathoverflow.net/questions/4200/flood-of-new-users $\endgroup$– YCorCommented May 31, 2019 at 13:57
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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.
