Computing the sheaf of 1-forms on a toric variety Consider projective space $P^{2}$ and its corresponding fan. We have the affine opens defined by $U_{\sigma_{0}} = Spec(\mathbb{C}[x,y])$, $U_{\sigma_{1}} = Spec(\mathbb{C}[x^{-1},x^{-1}y])$ and $U_{\sigma_{2}} = Spec(\mathbb{C}[xy^{-1},y^{-1}])$ cf (Example 3.1.9 Cox, Little, and Schenck). 
Given these affine opens, how would one compute the sheaves $\Omega^{1}_{\mathbb{P}^{2}}(U_{\sigma_{i}})$ and the transition functions on the intersections $U_{\sigma_{i}} \cap U_{\sigma_{j}}$? (cf Exercise 8.2.4 Cox, Little, and Schenck).
I understand the construction via using the transformation of the differential forms with the Jacobian matrix, but I'm not sure how to work using the above approach. My main issue is that I'm not sure how to compute $\Omega^{1}_{\mathbb{P}^{2}}(U_{\sigma_{i}})$.
Thanks
 A: Let's recall the definition of the module of differentials on a $\mathbb{C}$-algebra $R$: take the free $R$-module on the symbols $\{ dr \colon r \in R \}$ and mod out by the relations $d(r+s ) = dr + ds$, $d(rs) = sdr + r ds$, and $da = 0$ for $a \in \mathbb{C}$. Then, the sheaf $\Omega_{R/\mathbb{C}}^1$ on $\textrm{Spec}(R)$ is the "tilde" of this module. A useful fact for computations (which will be used below) is the following: it suffices to take the free module on the symbols $\{ dr_i \}_{i \in I}$, where $\{ r_i \in R \colon i \in I \}$ is a generating set (and then mod out by the appropriate relations). 
On each piece of the given open affine cover of $\mathbb{P}^2_{\mathbb{C}}$, the modules of differentials are
$$
\Omega_{\mathbb{P}^1}^1 (U_{\sigma_0}) = \mathbb{C}[x,y] \cdot dx \oplus \mathbb{C}[x,y] \cdot dy
$$
\begin{align*}
\Omega_{\mathbb{P}^1}^1 (U_{\sigma_1}) 
&= \mathbb{C}[x^{-1},x^{-1} y] \cdot d(x^{-1}) \oplus \mathbb{C}[x^{-1}, x^{-1} y ] \cdot d(x^{-1} y)\\
&= \mathbb{C}[x^{-1}, x^{-1}y] \cdot x^{-2} dx \oplus \mathbb{C}[x^{-1}, x^{-1} y ] \cdot yx^{-2} dx \oplus \mathbb{C}[x^{-1}, x^{-1}y ] \cdot x^{-1} dy
\end{align*}
Knowing this, can you now compute $\Omega_{\mathbb{P}^1}^1(U_{\sigma_2})$? Edit: for the sake of completeness, the module of differentials on $U_{\sigma_2}$ is
\begin{align*}
\Omega_{\mathbb{P}^1}^1(U_{\sigma_2}) 
&= \mathbb{C}[xy^{-1}, y^{-1}] \cdot d(xy^{-1}) \oplus \mathbb{C}[xy^{-1}, y^{-1}] \cdot d(y^{-1})\\
&= \mathbb{C}[xy^{-1}, y^{-1}] \cdot y^{-1} dx \oplus \mathbb{C}[xy^{-1}, y^{-1}] \cdot xy^{-1} dy \oplus \mathbb{C}[xy^{-1}, y^{-1}] \cdot y^{-2} dy.
\end{align*}
Furthermore, because of how we presented the modules $\Omega_{\mathbb{P}^1}^1(U_{\sigma_0})$ and $\Omega_{\mathbb{P}^1}^1 (U_{\sigma_1}) $ above, we have already calculated the transition functions! For example, on the intersection $U_{\sigma_0} \cap U_{\sigma_1} = \textrm{Spec}\left( \mathbb{C}[x,x^{-1},y,x^{-1}y] \right)$, we need to determine how to write $d(x^{-1})$ and $d(x^{-1} y)$ in terms of $dx$ and $dy$, respectively. The rules defining the module of differentials tell us that
$$
d(x^{-1}) = x^{-2} dx
$$
and
$$
d(x^{-1} y) = x^{-2} y dx + x^{-1} dy,
$$
and so we have our transition functions on $U_{\sigma_0} \cap U_{\sigma_1}$. The same process can be repeated on the other two intersections $U_{\sigma_0} \cap U_{\sigma_2}$ and $U_{\sigma_1} \cap U_{\sigma_2}$. 
