[Line Bundles, Invertible Sheaves and Sheaf Cohomology I & II] Line Bundles & Invertible Sheaves

In this short note, we will present the connection between line bundles and invertible sheaves. In the first part of this series, we already mentioned about rational sections, and see how they are connected to the Picard group. For our purpose, we need the notion of regular section. We always fix $X$ a smooth projective curve, but one can see that it is used nowhere in this note, and hence, is true for any scheme.

1. Regular sections of line bundles. 

Definition 1.1. Let $(L,\pi)$ be a line bundle over $X$, and $U\subset X$ an open set. A regular section of $L$ over $U$ is a function $s:U\to L$ such that:
1. $\pi\circ s=id_U$
2. For any line bundle chart $(V,\phi)$ of $L$, the map $pr_1\circ \phi\circ s: U\cap V\to \mathbb{C}$ is a regular function in $U\cap V$.

We will analyze the definition and see how it relates to the definition of line bundles. First, one can see for any regular section $s$ of $L$ over $U$, one can see from the definition that $s(p)\in \pi^{-1}(U)$, for all $p\in U$. Let $(U_i,\pi_i)$ be line bundle charts of $L$. Then $\{U_i\}$ is an open cover of $X$, and hence, for any $p\in U$, there exists $U_i$ such that $p\in U_i$. Then one can see $s(p)\in \pi^{-1}(U_i)$, which is in bijection with $\mathbb{C}\times U_i$. Let $\pi_i(s(p))=(f_s(p),q)\in\mathbb{C}\times U_i$. But then due to the first requirement of the definition above, and the definition of line bundle charts, we have $p=\pi\circ s(p)= pr_2\circ \pi_i(s(p))=pr_2(f_s(p),q)=q$. That means, $q=p$, and $s(p)$ is of the form $(f_s(p),p)$. The second requirement of Definition 1.1. implies that $f_s(p)$ is a regular function on $U\cap U_i$.

For any line bundle $L$, if we denote $\mathscr{O}\{L\}(U)$ the set of all regular sections of $L$ over $U$, then for any $s_1,s_2\in \mathscr{O}\{L\}(U)$, we can define $(s_1+s_2)(p):=(f_{s_1}(p)+f_{s_2})(p),p)$. It can be seen that $s_1+s_2\in  \mathscr{O}\{L\}(U)$, and this gives $ \mathscr{O}\{L\}(U)$ the abelian group structure. Moreover, for any $f\in  \mathscr{O}(U)$, we define $(fs)(p):=(f(p)f_s(p),p)$, and also $(fs) \in \mathscr{O}\{L\}(U)$. This gives $ \mathscr{O}\{L\}(U)$ the $ \mathscr{O}(U)$-module structure. Even more interesting, we have

Lemma 1.2. $\mathscr{O}\{L\}$ is a sheaf of $\mathscr{O}$-module.

Proof. It can be seen that $\mathscr{O}\{L\}$ is a presheaf. We now check the gluing condition. Suppose, $U\subset X$ is an open subset, and $s_i$ are regular section of $L$ over $U_i$, such that $s_{U_i\cap U_j}=s_{U_i\cap U_j}$. We then have a well-defined map $s: U\to L$ such that $s_{U_i}=s_i$ and obviously it satisfies the first condition of Definition 1.1. Moreover, assume that $(V,\phi)$ is any line bundle chart of $L$, then $pr_1\circ \phi\circ s_i: U_i\cap V\to \mathbb{C}$ is in $\mathscr{O}(V\cap U_i)$, and they agree on the intersections. This implies we can glue them all to obtain a unique regular function $f\in \mathscr{O}(V\cap U)$. And it is obvious from our construction that $f=pr_1\circ\phi\circ s$. (Q.E.D)

We then conclude this section by

Proposition 1.4. $\mathscr{O}\{L\}$ is an invertible sheaf.

Proof. We need to find an open cover $U_i$ of $X$ such that $\mathscr{O}\{L\}_{U_i}$ is isomorphic to $\mathscr{O}_{U_i}$. Let $(U_i,\pi_i)$ be line bundle charts of $L$, we just need to prove that for all $V\subset U_i$, $\mathscr{O}\{L\}(V)$ is isomorphic to $\mathscr{O}(V)$. But this can be done via the second requirement of Definition 1.1. Let $f_s:=pr_1\circ \pi_i\circ s$, i.e. $f_s$ is constructed as above. Then it can be seen that $f_s$ is regular on $\mathscr{O}(V)$. Conversely, for any $f\in \mathscr{O}(V)$, we can construct $s_f(p)=\pi_i^{-1}(f(p),p)\in \pi^{-1}(U_i)$. Then the two maps $s\mapsto f_s$ and $f\mapsto s_f$ are inverses of each other, since $s_{f_s}(p)=\pi_i^{-1}(f_s(p),p)=s(p)$, and $f_{s_f}(p)=pr_1\circ \pi_i\circ s_f(p)=pr_1\circ\pi_i\circ\pi_i^{-1}(f(p),p)=f(p)$. And the two maps can be extended linearly to $\mathscr{O}(V)$-module. This yields the isomorphism between the two sheaves. (Q.E.D)

2. Cocycle conditions for regular sections. We again meet the cocycle conditions for regular sections. It will give the simpler description of regular sections. If we begin with a regular section $s$ on $U$, then we can see from the first section that there always exists a corresponding regular function $f_i$ in $\mathscr{O}(U\cap U_i)$. And due to the compatible condition for line bundle charts, one can see on $U\cap U_j$, we have $f_i=t_{ij}f_j$, where $t_{ij}$ is the transition function.

Conversely, if we begin with $f_i\in \mathscr{O}(U_i)$, such that $f_i=t_{ij}f_j$. Then one can define for $p\in U_i$, $s(p)=\pi_i^{-1}(f_i(p),p)$.  Note that it is a well-defined, because if $p\in U_i\cap U_j$, we have $\pi_i^{-1}(f_i(p),p)=\pi_j^{-1}(f_j(p),p)$, since $\pi_j\circ\pi_i^{-1}(f_i(p),p)=(t_{ij}(p)f_i(p), p)=(f_j(p),p)$. Hence, to construct regular section, we can begin with any open subset $U\subset X$, and $f_i\in \mathscr{O}(U\cap U_i)$ such that $f_i=t_{ij}f_j$ in $U\cap U_i\cap U_j$. Then $s(p):=\pi_i^{-1}(f_i(p),p)$ is a regular section of $L$ over $U$.

We conclude this note by the following

Proposition 2.1. Let $(L,\pi)$ be a line bundle on $X$, $(U_i,\pi_i)$ line bundle charts with $t_{ij}$ the transition functions from $U_i$ to $U_j$, and $U\subset X$ is open. Then given a regular section of $L$ over $U$ is equivalent to be given a tuple of regular function $f_i\in \mathscr{O}(U\cap U_i)$, such that $f_i=t_{ij}f_j$ on $U\cap U_i\cap U_j$.