Recall that in previous notes, we have proved the following three things are in fact the same for smooth projective curves: the Picard group, invertible sheaves modulo isomorphism and $H^1(X,\mathscr{O}^*)$. In this note, we will see the bijection between $H^1(X,\mathscr{O}^*)$ and the sets of line bundles modulo isomorphism. $X$ here can be any projective variety. We do not actually prove anything in this note, all come from "right" constructions.
Definition 1. Let $(L_1,\pi_1)$ and $(L_2,\pi_2)$ be two line bundles on $X$. A homomorphism from $(L_1,\pi_1)$ to $(L_2,\pi_2)$ is a map $\alpha: L_1\to L_2$ such that
1. $\pi_2\circ\alpha=\pi_1$.
2. For any line bundle chart $(U,\phi)$ of $L_1$ and $(V,\psi)$ of $L_2$, $\psi\circ\phi^{-1}|_{k\times (U\cap V)}$ sends $(v,p)$ to $(s(p)v,p)$, where $s\in\mathscr{O}(U\cap V)$.
If there exists $\beta$ a line bundle homomorphism from $L_2$ to $L_1$, and $\alpha,\beta$ are inverses of each other, then $\alpha$ is called the isomorphism between line bundles.
The following constructions are very important.
Construction 2. Let $(L,\pi)$ be a line bundle on $X$, with charts $(U_i,\phi_i)$ and $s_i\in \mathscr{O}^*(U_i)$, i.e. $s_i$ is nowhere zero regular function on $U_i$. We then have the bijective map $S_i: k\times U_i\to k\times U_i$ sending $(v,p)$ to $(s_i(p)v,p)$. Define $\theta_i: \pi^{-1}(U_i)\to k\times U_i$ the composition of $S_i\circ\phi_i$. We have $\theta_i$ is the bijection from $\pi^{-1}(U_i)$ to $k\times U_i$.
Lemma 3. $(U_i,\theta_i)$ defines a line bundle structure for $(L,\pi)$, with transition functions are $(s_j/s_i)t_{ij}$, where $(t_{ij})$ are transition functions of $(U_i,\phi_i)$ and two line bundle structures of $L$ is in fact isomorphic.
Proof. We first see that $(U_i,\theta_i)$ are line bundle charts for $(L,\pi)$. The only thing left is the compatible condition. We have
$$\theta_j\theta_i^{-1}(v,p)=S_j\circ\phi_j\circ\phi_i^{-1}\circ S_i^{-1}(v,p)=(\frac{s_j}{s_i}(p)t_{ij}(p)v,p)$$
where $t_{ij}$ are transition functions for the first line bundle structure of $L$. Then one can see $(s_j/s_i)t_{ij}\in \mathscr{O}^*(U_i\cap U_j)$. Hence, $(U_i,\theta_i)$ is also a line bundle structure for $L$, with transition functions are $(s_j/s_i)t_{ij}$.
Let us check the two line bundle structure is isomorphic. We first have $id:L\to L$ satisfying the first condition. For the second condition, we have $\theta_j\circ\phi_i^{-1}(v,p)=S_j\circ\phi_j\circ\phi_i^{-1}(v,p)=(s_jt_{ij}(p)v,p)$, and $s_jt_{ij}\in \mathscr{O}^*(U_i\cap U_j)$. Hence, $id$ is a line bundle homomorphism, with its inverse is itself. Equivalently, we obtain the isomorphism between two line bundle structures. (Q.E.D)
Construction 4. Let $s_i\in \mathscr{O}^*(U_i)$, where $\{U_i\}$ are open cover of $X$, and $t_{ij}:=s_j/s_i\in \mathscr{O}^*(U_\cap U_j)$. We can construct a line bundle $L$ with transition functions $t_{ij}$ as follows. Let $L:=k\times X$, and $\pi:L\to X$ the projection. And $\pi^{-1}(U_i)=k\times U_i$. We then define line bundle charts $\phi_i:k\times U_i\to k\times U_i$ sending $(v,p)$ to $(s_i(p)v,p)$. And $\phi_j\circ\phi_i^{-1}(v,p)=(t_{ij}(p)v,p)$, which satisfies the compatible condition. Hence, $L$ is a line bundle on $X$. We will prove that, in fact, $L$ is a unique line bundle having $t_{ij}$ as transition functions (up to isomorphism).
Lemma 5. If $(L',\pi')$ is a line bundle on $X$, with charts $(U_i,\theta_i)$ with $t_{ij}$ the transition functions as defined above. Then $L\cong L'$.
Proof. We first define the map $\alpha: L\to L'$. The most natural one is the composition of $\pi^{-1}(U_i)\xrightarrow{\phi_i}k\times U_i\xrightarrow{\theta_i^{-1}}\pi'^{-1}(U_i)$, which sends $x\in \pi^{-1}(U_i)$ to $\theta_i^{-1}\circ\phi_i(x)\in \pi'^{-1}(U_i)$. This map is well-defined and bijective, since if $x\in \pi^{-1}(U_i)\cap \pi^{-1}(U_j)$, then $\theta_i^{-1}\circ\phi_i(x)=\theta_j^{-1}\circ\phi_j(x)$ (equivalently, $\theta_j\circ\theta_i^{-1}(v,p)=\phi_j\circ\phi_i^{-1}(v,p)=(t_{ij}(p)v,p)$, due to the our assumption).
From this, on $U_i\cap U_j$, we have $\theta_j\circ\alpha\circ\phi_i^{-1}(v,p)=\theta_j\circ\theta_i^{-1}\circ\phi_i\circ\phi_i^{-1}(v,p)=(t_{ij}(p)v,p)$, and $t_{ij}\in \mathscr{O}^*(U_i\cap U_j)$. This yields $\alpha^{-1}$ is also a line bundle homomorphism, and hence $L\cong L'$. (Q.E.D)
In fact, what we have proved is that from a cocycle $t_{ij}$ in $H^1(X,\mathscr{O}^*)$, we can define a unique line bundle such that $t_{ij}$ are transitions functions of a line bundle. This shows that the map from the quotient set $LB(X)$ (which is the set of all line bundles over $X$ modulo isomorphism) to $H^1(X,\mathscr{O}^*)$ is well-defined, and surjective.
Lemma 6. The map from $LB(X)$ to $H^1(X,\mathscr{O}^*)$ is actually injective.
Proof. If we have two line bundles $(L_1,\pi_1)$, with charts $(U_i,\phi_i)$ and $(L_2,\pi_2)$ with charts $(U_i,\psi_i)$ (note that we can refine our open sets, so that the two line bundles have the same supports) that map to the same cocycle in $H^1(X,\mathscr{O}^*)$, i.e. our two line bundles have the transition functions satisfying $t_{ij}^{(2)}=s_j/s_it_{ij}^{(1)}$, where $s_i\in \mathscr{O}^*(U_i),s_j\in \mathscr{O}^*(U_j)$. We will prove that $L_1\cong L_2$.
Now, via Construction 1, we obtain the line bundle $L_1$ with isomorphic structure, which has transition function $s_j/s_it_{ij}^{(1)}$, which are the same with transition functions of $L_2$. Hence, by Construction 4, $L_1\cong L_2$. (Q.E.D)
We then obtain the bijection between $LB(X)$ and $H^1(X,\mathscr{O}^*)$.
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