Sheaf of differetials on $\mathbb{P}^n_A$

Question

I am not clear how to display the sheaf of differetials $\Omega_{X/A}$ on $X=\mathbb{P}^n_A$ explicitly,

What is its gobal section $\Omega_{X/A}(X)$ and section on the complement of the hyperplane $T_0=0$, $\Omega_{X/A}(D_+(T_0))$, and its stalk ${\Omega_{X/A}}_x$?

I am reading a proof that the sheaf of differetials $\Omega_{X/A}$ on $X=\mathbb{P}^1_A$ isomorphic to $\mathcal{O}_X(-2)$.(Q.Liu, Algebraic geometry and arithematical curves, P217, Example 1.22)

It is argued that on $D_+(T_0)\cap D_+(T_1)$, we have $T_0^2d(T_1/T_0)=T_1^2d(T_0/T_1)$. This induces a global section of $\mathcal{O}_X(2)\otimes\Omega_{X/A}(X)$. It can be checked isomorphic to $\mathcal{O}_X$. So $\Omega_{X/A}$ is isomorphic to $\mathcal{O}_X(-2)$.

I am not clear how to check $\mathcal{O}_X(2)\otimes\Omega_{X/A}(X) \cong \mathcal{O}_X$ , on the stalk?

And does it hold or not for higher dimension projective spaces?

Answer 1

I'm going to first answer your second question which is establishing the isomorphism

$$\mathcal{O}_X(2)\otimes\Omega_{X}(X) \cong \mathcal{O}_X$$

for $X=\mathbb{P}^1.$

For constructing the cotangent sheaf, we start with the module of relative differentials $\Omega_A$ and then by definition, $\Omega_X$ will be the coherent sheaf associated to $\Omega_A.$ Let's find out what $\Omega_A$ is?

If you review the construction of the module of differentials, you'll see that we start with a free module generated by $dT_0, dT_1$ i.e. $AdT_0 \oplus AdT_1,$ then we mode out by relations, which in our case, there is only one relation, $T_0^2d(T_1/T_0)=T_1^2d(T_0/T_1)$ in the overlap $D_+(T_0) \cap D_+(T_1)$ which after differentiation leads to $T_0dT_1=T_1dT_0.$

Therefore,

$$\Omega_A=AdT_0 \oplus AdT_1/(T_0dT_1-T_1dT_0).$$

It's enough to establish the isomorphism on stalks, so let $p \in D_+(T_0)$ and $\mathfrak{p}$ be the prime ideal of our point. We claim that

$$A_\mathfrak{p}dT_0 \oplus A_\mathfrak{p}dT_1/(T_0dT_1-T_1dT_0) \cong A_\mathfrak{p}$$

Proof: Take an element $fdT_0+gdT_1 \in A_\mathfrak{p}dT_0 \oplus A_\mathfrak{p}dT_1/(T_0dT_1-T_1dT_0)$ and send it to $fT_0+gT_1 \in A_{\mathfrak{p}}.$ Conversely, take an element $h \in A_{\mathfrak{p}}$ and send it to $(h/T_0)dT_0 \in A_\mathfrak{p}dT_0 \oplus A_\mathfrak{p}dT_1/(T_0dT_1-T_1dT_0).$ It's easy to show that it's in fact an isomorphism.

The case for localizing at a point $q \in D_+(T_1)$ is quite similar. I'll leave it to you to fill the gaps. Also, note that we're using the relation $dT_0/T_0=dT_1/T_1$ on the overlap $D_+(T_0) \cap D_+(T_1)$ to have a consistent morphism.

You can easily generalize this isomorphism and show that $\Lambda^n \Omega_{\mathbb{P}^n} \cong \mathcal{O}_{\mathbb{P}^n}(-n-1).$

By now, you should be able to somehow answer your first question.

Answer 2

For the general case: If $\mathcal{E}$ is some quasi-coherent sheaf on a scheme $S$ and $p : \mathbb{P}(\mathcal{E}) \to S$ is the associated projective bundle, then we have an exact sequence $$0 \to \Omega^1_{\mathbb{P}(\mathcal{E})/S} \to p^*(\mathcal{E})(-1) \to \mathcal{O}_{\mathbb{P}(\mathcal{E})} \to 0.$$ This is known (at least in the well-known special case where $\mathcal{E}=\mathcal{O}_S^{n+1}$) as the Euler sequence.