Smooth-locus restriction properties

PDF p. 9: Functions on the smooth locus; PDF p. 10: Lemmas 5.1 and 5.2; PDF p. 15: Singular locus

Assume the Axiom of Choice. Let $k$ be an algebraically closed field and $X$ a finite-type $k$-scheme. If $X$ is reduced, its smooth locus $U$ is open and meets every irreducible component densely. In particular, this applies to the reduced seminormal schemes in the proof of Lemma 5.1 of the article. If $X$ is normal, then $U$ is dense, $X\setminus U$ has codimension at least two in every irreducible component, and restriction induces isomorphisms $\mathrm{\Gamma }(X,{\mathcal{O}}_X)\cong \mathrm{\Gamma }(U,{\mathcal{O}}_U)$ and $\mathrm{\Gamma }(X,{\mathcal{O}}_X{)}^{\times }\cong \mathrm{\Gamma }(U,{\mathcal{O}}_U{)}^{\times }$, as well as an injection $\mathrm{Pic}(X)\hookrightarrow \mathrm{Pic}(U)$.

Here algebraically closed means that every nonconstant polynomial over $k$ has a root in $k$. Finite type means that $X$ has a finite affine cover by spectra of finitely generated $k$-algebras. Reduced means that its local rings have no nonzero nilpotents; normal means that its local rings are integrally closed domains. The smooth locus is the set of points at which the structure morphism $X\to \mathrm{Spec}k$ is smooth. Codimension of a point in an irreducible component is the supremum of the lengths of strict chains of irreducible closed subsets from that component to the closure of the point; the codimension of a subset is the infimum of the codimensions of its points, with the empty subset assigned $+\mathrm{\infty }$. A global unit is a global function with a multiplicative inverse. The Picard group is the group of isomorphism classes of line bundles, meaning locally free ${\mathcal{O}}_X$-modules of rank one, under tensor product. Restriction is induced by pulling back functions or line bundles along $U\hookrightarrow X$.