Lemma 54.3.2. Let $(A, \mathfrak m, \kappa )$ be a regular local ring of dimension $2$. Let $f : X \to S = \mathop{\mathrm{Spec}}(A)$ be the blowing up of $A$ in $\mathfrak m$. Then $X$ is an irreducible regular scheme.

Proof. Observe that $X$ is integral by Divisors, Lemma 31.32.9 and Algebra, Lemma 10.106.2. To see $X$ is regular it suffices to check that $\mathcal{O}_{X, x}$ is regular for closed points $x \in X$, see Properties, Lemma 28.9.2. Let $x \in X$ be a closed point. Since $f$ is proper $x$ maps to $\mathfrak m$, i.e., $x$ is a point of the exceptional divisor $E$. Then $E$ is an effective Cartier divisor and $E \cong \mathbf{P}^1_\kappa$. Thus if $g \in \mathfrak m_ x \subset \mathcal{O}_{X, x}$ is a local equation for $E$, then $\mathcal{O}_{X, x}/(g) \cong \mathcal{O}_{\mathbf{P}^1_\kappa , x}$. Since $\mathbf{P}^1_\kappa$ is covered by two affine opens which are the spectrum of a polynomial ring over $\kappa$, we see that $\mathcal{O}_{\mathbf{P}^1_\kappa , x}$ is regular by Algebra, Lemma 10.114.1. We conclude by Algebra, Lemma 10.106.7. $\square$

Comment #4247 by Dario Weißmann on

the local equation for $E$ has the same name as the morphism $f:X\to S$. Maybe call it $g$ instead?

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