Line bundles over $\Bbb P^1_k$ I am trying to understand the line bundle $O(1)$ over $\Bbb P^1_k$ and why 

$$ O(1)^{\otimes n} = O(n)$$ from Vakil's notes, p398, line 6. 

His explanation is rather long, so I took a screen shot. 


I don't understand Vakil's explanation of why we have the equality because of the transition functions. 

 A: The whole point here is that if we take two invertible sheaves $\scr L$ and $\scr M$ on $X$ and an open cover $\{U_i\}$ of $X$ such that $\mathscr L$ and $\scr M$ are both trivializable, so that $\scr L$ and $\scr M$ are given by transition functions (satisfying the cocyle conditions) over the intersections, say by $f_{ij}\in\mathscr O_X(U_{ij})^\times$ and $g_{ij}\in\mathscr O_X(U_{ij})^\times$ respectively (I am writing $U_{ij}:=U_i\cap U_j$), then $\mathscr L\otimes\scr M$ is also trivializable on the cover $\{U_i\}$ and the associated transition functions for the tensor product on this cover are exactly given by $f_{ij}g_{ij}$ over $U_{ij}$.
How to see this? Well let's recall quickly where the "transition functions" are coming from; for each $i$ we have isomorphisms $\varphi_i:\mathscr L|_{U_i}\to\mathscr O_X|_{U_i}$, and then over $U_{ij}$ we have an isomorphism
$$\mathscr O_X|_{U_{ij}}\overset{\varphi_i^{-1}|_{U_{ij}}}\longrightarrow\mathscr L|_{U_{ij}}\overset{\varphi_j|_{U_{ij}}}\longrightarrow\mathscr O_X|_{U_{ij}}$$
but any such isomorphism is determined by a global section of $\mathscr O_X|_{U_{ij}}$, given by the image of $1\in\mathscr O_{X}(U_{ij})$ under the map on sections of $U_{ij}$, and this is exactly the element $f_{ij}$ we are calling the transition function (sorry for the drawn out details if this is all obvious to you); similarly we have $g_{ij}$ coming from $\psi_j\circ\psi_i^{-1}$ (actually the restrictions to $U_{ij}$ as we wrote more carefully above), where $\psi_i:\mathscr M|_{U_i}\to\mathscr O_X|_{U_i}$ are some local trivializations for $\mathscr M$.
Now notice we have a sequence of isomorphisms
$$(\mathscr L\otimes\mathscr M)|_{U_i}\longrightarrow\mathscr L|_{U_i}\otimes\mathscr M|_{U_i}\overset{\varphi_i\otimes\psi_i}\longrightarrow\mathscr O_X|_{U_i}\otimes\mathscr O_X|_{U_i}\longrightarrow\mathscr O_X|_{U_i}$$
(the composition of which we will call $\rho_i$) where the last map is just multiplication, and this gives us explicit trivializations of $\mathscr L\otimes\mathscr M$ on the cover $\{U_i\}$. Now that we know more precisely what the transitions exactly are from our discussion above, let's compute them with respect to our chosen trivializations $\rho_i$; the map $\rho_j\circ\rho_i^{-1}$ on sections over $U_{ij}$ is explicitly the composition
$$\mathscr O_X(U_{ij})\to\mathscr O_X(U_{ij})\otimes\mathscr O_X(U_{ij})\overset{\varphi_i^{-1}\otimes\psi_i^{-1}}\longrightarrow\mathscr L(U_{ij})\otimes\mathscr M(U_{ij})\overset{\varphi_j\otimes\psi_j}\longrightarrow\mathscr O_X(U_{ij})\otimes\mathscr O_X(U_{ij})\to\mathscr O_X(U_{ij})$$
and then if we chase down where $1\in\mathscr O_X(U_{ij})$ goes under this map we find
$$1\mapsto 1\otimes 1\mapsto \varphi_i^{-1}(1)\otimes\psi_i^{-1}(1)\mapsto(\varphi_j\circ\varphi_i^{-1})(1)\otimes(\psi_j\circ\psi_i^{-1})(1)=f_{ij}\otimes g_{ij}\mapsto f_{ij}g_{ij}$$
proving $f_{ij}g_{ij}$ indeed are the transition functions of the tensor product over this cover.
Hope this helps; when I was learning this stuff I was furious there were tons of details like this nobody was being precise about, so I tried to be as explicit as I could.
