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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)