Let $k$ be a field. We define the scheme $X = \mathbb{P}^1_k$ to be the gluing of the affine schemes $\text{spec}(k[T])$ and $\text{spec}(k[U])$ via the isomorphism \begin{align*} \phi: k[T,T^{-1}] &\to k[U,U^{-1}] \\ T &\mapsto \frac{1}{U} \end{align*} We want to compute the canonical sheaf $\Omega_{X \mid k}$. We have an explicit description of the canonical bundle on $\text{spec}(k[T])$ via $(\Omega_{k[T] \mid k})^\sim$, and we have $\Omega_{k[T] \mid k} = k[T] \cdot dT$, and similarly for $\text{spec}(k[U])$. Now, I want to glue these to an $\mathcal{O}_X$-module on $X$. We have that $\Omega_{k[T,T^{-1}] \mid k} = k[T,T^{-1}] \cdot dT$, and $\Omega_{k[U,U^{-1}] \mid k} = k[U,U^{-1}] \cdot dU$.
It seems to me that we should glue $\Omega_{k[T,T^{-1}]}$ and $\Omega_{k[U,U^{-1}]}$ via an isomorphism of $k[T,T^{-1}]$-modules, where we view $\Omega_{k[U,U^{-1}]}$ as an $k[T,T^{-1}]$-module via $\phi$. Any such morphism is given by mapping $dT \mapsto \pm U^{\alpha}dU$, where $\alpha$ is some integer.
Furthermore, we have that $\Omega_{k[T,T^{-1}]\mid k} \cong k[T,T^{-1}]$ via mapping $dT$ to any $\pm T^{\beta}$, for any integer $\beta$ and $\Omega_{k[U,U^{-1}]\mid k} \cong k[U,U^{-1}]$ via mapping $dU$ to any $\pm U^{\gamma}$, for any integer $\gamma$.
It seems to me that all of these morphisms are compatible given that $\alpha + \beta + \gamma = 0$, and that this describes $\Omega_{X \mid }$ as a Line bundle on $X$.
Questions:
- Can I choose whatever $\alpha$ and sign that I want? To me the choice $\alpha = -2$ seems the most natural since if we identify $U$ with $1/T$ we get $dT = d\frac{1}{U} = -\frac{dU}{U^2} = -T^2dU$
- I think that I should be able to get a Cartier divisor by specifying the data $\{(\text{spec}(k[T,T^{-1}],f),(\text{spec}(k[U,U^{-1}],g)\}$ for some $f$ and $g$, and these $f$ and $g$ should be connected to the two last iomorphisms above, but what exactly are $f$ and $g$ and their relation? Will a different choice of $\alpha$, $\beta$, $\gamma$ result in equivalent Cartier Divisors?
As I have issues with the identifications, I would appreciate if the answer actually treats those with care.