The Stacks project

Lemma 54.3.6. 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 $\Omega _{X/S} = i_*\Omega _{E/\kappa }$, where $i : E \to X$ is the immersion of the exceptional divisor.

Proof. Writing $\mathbf{P}^1 = \mathbf{P}^1_ S$, let $r : X \to \mathbf{P}^1$ be as in Lemma 54.3.1. Then we have an exact sequence

\[ \mathcal{C}_{X/\mathbf{P}^1} \to r^*\Omega _{\mathbf{P}^1/S} \to \Omega _{X/S} \to 0 \]

see Morphisms, Lemma 29.32.15. Since $\Omega _{\mathbf{P}^1/S}|_ E = \Omega _{E/\kappa }$ by Morphisms, Lemma 29.32.10 it suffices to see that the first arrow defines a surjection onto the kernel of the canonical map $r^*\Omega _{\mathbf{P}^1/S} \to i_*\Omega _{E/\kappa }$. This we can do locally. With notation as in the proof of Lemma 54.3.1 on an affine open of $X$ the morphism $f$ corresponds to the ring map

\[ A \to A[t]/(xt - y) \]

where $x, y \in \mathfrak m$ are generators. Thus $\text{d}(xt - y) = x\text{d}t$ and $y\text{d}t = t \cdot x \text{d}t$ which proves what we want. $\square$

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