Computing canonical sheaf of the total space of line bundle $\mathcal{O}_{\mathbb{P}^n}(m)$ over $k$

Question

Let $U$ be the total space of line bundle whose sheaf of sections is $\mathcal{O}_{\mathbb{P}^n}(m)$. Let $\pi:U\rightarrow \mathbb{P}^n$ be the natural projection. I want to show the following sequence is exact:

$0\rightarrow \pi^*\Omega_{\mathbb{P}^n}\rightarrow \Omega_{U/k}\rightarrow \pi^*\mathcal{O}_{\mathbb{P}^n}(-m)\rightarrow 0$.

First, I will restrict them to each $U_i=Spec k[\frac{x_0}{x_i},\cdots, \frac{x_n}{x_i}][y_i]$

$0\rightarrow \pi^*\Omega_{\mathbb{P}^n}|_{U_i}\rightarrow \Omega_{U/k}|_{U_i}\rightarrow \pi^*\mathcal{O}_{\mathbb{P}^n}(-m)|_{U_i}\rightarrow 0$.

Then I compute $\Gamma(U_i,\pi^*\Omega_{\mathbb{P}^n}|_{U_i})=\Omega_{k[\frac{x_0}{x_i},\cdots, \frac{x_n}{x_i}]/k}\otimes_{k[\frac{x_0}{x_i},\cdots, \frac{x_n}{x_i}]} k[\frac{x_0}{x_i},\cdots, \frac{x_n}{x_i}][y_i]$

$\Gamma(U_i,\Omega_U|_{U_i})=\Omega_{k[\frac{x_0}{x_i},\cdots, \frac{x_n}{x_i}][y_i]/k}$

$\Gamma(U_i,\pi^*\mathcal{O}(-m)|_{U_i})=x_i^{-m}k[\frac{x_0}{x_i},\cdots, \frac{x_n}{x_i}]\otimes_{k[\frac{x_0}{x_i},\cdots, \frac{x_n}{x_i}]}k[\frac{x_0}{x_i},\cdots, \frac{x_n}{x_i}][y_i]$.

But I don't know how to proceed from there.

Is there a way to see the sequence is exact without restricting to an open cover?

Answer 1

For any smooth morphism $\pi \colon X \to Y$ there is an exact sequence $$ 0 \to \pi^*\Omega_Y \to \Omega_X \to \Omega_{X/Y} \to 0. $$ In your case it remains to note that $$ \Omega_{U/\mathbb{P}^n} \cong \pi^*\mathcal{O}(-m) $$ because the morphism is the total space of the line bundle $\mathcal{O}(m)$.

Answer 2

Following Sasha's answer, the idea is to map $dy_i$ locally to $(\frac{1}{x_i})^m$ and check that on the intersection, everything matches.