Singularity is a local condition

Question

I'm reading Hartshorne's Algebraic Geometry, section I.$5$, where he defines singularity as this: $P\in X$ is said nonsingular if $\mathcal{O}_{P, X}$ is a regular local ring. (for $X$ any quasi-projective variety)

I've noticed that he implicitly assumes that if $P\in X$ is nonsingular and $U\subset X$ is an open subset containing $P$, then $P\in U$ is nonsingular.

I only know that $\mathcal{O}_{P, X}\subset \mathcal{O}_{P, U}$ and I can't justify $\mathcal{O}_{P, X}$ regular $\Rightarrow \mathcal{O}_{P, U}$ regular.

Answer 1

Stalks at a point are a local construction, i.e. there is a canonical isomorphism $\mathscr{O}_{P,X} \simeq \mathscr{O}_{P,U}$. (For the map from left to right, for a section $f \in \mathscr{O}_X(V)$ with $P \in V \subseteq X$, map it to $f |_{U \cap V} \in \mathscr{O}_X(U \cap V)$ and from there to $\mathscr{O}_{P, U}$. For the map from left to right, for a section $f \in \mathscr{O}_X(V)$ with $P \in V \subseteq U$, then also $P \in V \subseteq U \subseteq X$ so $f$ is also directly in the direct limit defining $\mathscr{O}_{P, X}$. Now, just check the well-definedness and compositions of these maps.)

Alternatively, to factor this argument into slightly higher-level concepts:

• Prove the directed set of open sets $V$ with $P \in V \subseteq U$ (ordered by reverse inclusion) is cofinal in the directed set of open sets $V$ with $P \in V \subseteq X$.
• Prove that if $J \subseteq I$ is a cofinal subset of a directed set $I$, then there is a canonical isomorphism $\underset{\longrightarrow \\ j \in J}{\lim} X_j \simeq \underset{\longrightarrow \\ i \in I}{\lim} X_i$.

Answer 2

We have equality of the stalks $\mathcal O_{X,P}=\mathcal O_{U,P}$. This follows immediately from the characterization of the stalk as a direct limit over all open sets that contain $P$.