In this note, we will prove the isomorphism between $Pic(X)$ and the group of invertible sheaves modulo isomorphism. They are in fact very close to the first cohomology group. We then use this to prove that the four things: Picard group, Line bundles modulo isomorphism, Invertible sheaves modulo isomorphism, and the first cohomology group are actually the same for smooth projective curves.
1. Isomorphism between sheaves. In algebra, it is never enough to define an algebraic structure without maps between objects. Let us first recall some basic knowledge of sheaf, and the reader may refer the earlier note on sheaf of $\mathscr{O}_X$-module, where $X$ is any scheme, and $\mathscr{O}_X$ is the sheaf of regular function on it. Recall that if $\mathscr{F}, \mathscr{G}$ are two sheaves of $\mathscr{O}_X$-module, then a morphism between them is denoted by $\phi: \mathscr{F}\to \mathscr{G}$ such that for all open subset $U$ of $X$, we have
1. $\phi_U: \mathscr{F}\to \mathscr{G}(U)$ is a $\mathscr{O}_X(U)$-module homomorphism.
2. For all $V\subset U$ is open, $p_V^U\circ \phi_U=\phi_V\circ p_V^U$, i.e. $\phi$ is compatible with the restriction.
We also recall that if $\phi$ is a morphism between two sheaves, then $\ker(\phi)$ is also a sheaf of $\mathscr{O}_X$-module. Because sheaf maps are similar to maps between $R$-modules, we need to define the notions of injective, onto, and isomorphism.
Definition 1.1. Let $\mathscr{F},\mathscr{G}$ be two sheaves, and $\phi$ is a morphism between them, then $\phi$ is injective if $\ker(\phi)(U)=0$ for all open subset $U$ of $X$. $\phi$ is onto if for all $p\in X$, and for all open subset $U$ of $X$ containing $p$, and for all $g\in \mathscr{G}(U)$, there exists open subset $V\subset U$ such that $p\in V$, and $f\in \mathscr{F}(V)$ such that $\phi_V(f_V)=g_V$. $\phi$ is isomorphism if $\phi$ is both injective and onto.
Note that the onto definition is not surjective map between $\mathscr{F}(U)$ and $\mathscr{G}(U)$ for all $U$, and it is actually related to the first cohomology group, which will be discussed later. But we have some similarities at hand.
Lemma 1.2. Let $\mathscr{F}, \mathscr{G}$ be two sheaves and $\phi$ is a morphism between them, then $\phi$ is an isomorphism iff it has an inverse sheaf map. Equivalently, $\phi$ is an isomorphism iff for all open subset $U\subset X$, the map $\phi_U: \mathscr{F}(U)\to \mathscr{G}(U)$ is an $\mathscr{O}$-module isomorphism.
We will next go to the definition of invertible sheaves on curves, and see how they relate to divisors on curves.
2. Invertible sheaves. In this section, we always denote $X$ a smooth projective curve, and $\mathscr{O}$ the sheaf of regular function on $X$.
Definition 2.1. Let $\mathscr{F}$ be a sheaf of $\mathscr{O}$-module on $X$, then $\mathscr{F}$ is called invertible if for all $p\in X$, there exists an open neighborhood $U$ of $p$, such that $\mathscr{F}_U\cong \mathscr{O}_U$ as sheaves of $\mathscr{O}_U$-module. This isomorphism $\phi$ between $\mathscr{O}_U$ and $\mathscr{F}_U$ is called trivialization of $\mathscr{F}$.
One can see that the definition of invertible sheaves above is equivalent to that there exists an open cover $\{U_i\}$ of $X$ such that $\mathscr{F}_{U_i}$ is isomorphic to $\mathscr{O}_{U_i}$ as sheaves of $\mathscr{O}_{U_i}$-module. Also, one can see that if $f_i$ is a generator of $\mathscr{F}(U_i)$ as $\mathscr{O}(U_i)$-module, then $f_{i_V}$ is also a generator of $\mathscr{F}(V)$ as $\mathscr{O}(V)$-module, for all open subset $V\subset U_i$. And the annihilator of $f_{i_V}$ is zero.
Conversely, if there exists $f_i\in \mathscr{F}(U_i)$ such that $f_{i_V}$ is a generator of $\mathscr{F}(V)$ as $\mathscr{O}(V)$-module for all open subset $V\subset U$, and the annihilator $f_{i_V}$ is zero, then $\mathscr{F}_{U_i}$ is isomorphic to $\mathscr{O}_{U_i}$ as $\mathscr{O}_{U_i}$-module. And hence, $\mathscr{F}$ is an invertible sheaf. We have proved
Proposition 2.2. Let $\mathscr{F}$ be a sheaf of $\mathscr{O}$-module, then $f$ is invertible iff there exists an open cover $\{U_i\}$ of $X$, and $f_i\in \mathscr{F}(U_i)$ such that for all open subset $V\subset U_i$, $f_{i_V}$ is the generator of $\mathscr{F}(V)$ as $\mathscr{O}(U_i)$-module, and the annihilator of $f_{i_V}$ is zero. Such $f_i$ is called local generator.
The most important example of invertible sheaves is of the form $\mathscr{O}[D]$, where $D\in Div(X)$ where $\mathscr{O}[D](U)=\{f\in k(X)^\times| div(f)+D\ge 0 \text{ on U}\}$. It can be seen that $\mathscr{O}[D]$ is a sheaf of $\mathscr{O}$-module. More precisely, we have
Proposition 2.3. Let $\mathscr{O}[D]$ be defined as above, then $\mathscr{O}[D]$ is an invertible sheaves, and for any $p\in X$, the local generator of $p$ is $z^{-D(p)}$, where $z$ is the local coordinate at $p$.
Proof. Let $U$ be an open neighborhood at $p$, defined by removing from $X$ all points $q$, that are zeros (except $p$) and poles of $z$, and $q$ in the support of $D$. Hence, $z$ is non-zero in $U\setminus \{p\}$. We will prove $z^{-D(p)}$ is the local generator of $\mathscr{O}[D](V)$, for all open subset $V\subset U$. It is quite clear, if we pick any $g\in \mathscr{O}[D](V)$ then $div(g)+D\ge 0$ in $V$ implies that for all point $q\in V, q\ne p$, and if $p\in U$ $ord_q(g)\ge 0$, and $ord_p(g)\ge -D(p)$. Hence, $r:=g/z^{-D(p)}$ is defined and regular on all $V$, i.e. $r\in \mathscr{O}(V)$. Also, assume that $rz^{-D(p)}=0$ on $V$, then $r=0$ in $V\setminus \{p\}$ (because $z$ is nowhere zero there), but the zeros of $r$ is closed in $V$, hence $r=0$ in $V$. This yields $z^{-D(p)}$ is the local generator of $\mathscr{O}[D](V)$ as $\mathscr{O}(V)$-module. Therefore, $\mathscr{O}[D]$ is an invertible sheaf. (Q.E.D)
Important Remark. Moreover, in Section 5, we will prove that, for any invertible sheaves $\mathscr{F}$, there always exists $D\in Div(X)$ such that $\mathscr{F}\cong \mathscr{O}[D]$ as $\mathscr{O}$-module. If this is proved, then for each isomorphism class of invertible sheaves, we can choose sheaf of the form $\mathscr{O}[D]$, and we will prove that it is in fact in bijection with $Pic(X)$. The bijection is actually an isomorphism of group, where the group law for invertible sheaves will be defined later.
3. Group law for the group of invertible sheaves on curves. We now turn to the construction section. Recall that we already define the sheafification of presheaf. But we need things more clear in specific context. We will now define the sheafification for tensor product and dual notion of sheaves.
If $\mathscr{F}, \mathscr{G}$ are two invertible sheaves of $\mathscr{O}$-module, then for an open subset $U\subset X$, we have $\mathscr{F}(U), \mathscr{G}(U)$ is $\mathscr{O}(U)$-module. And one can define the tensor product $\mathscr{F}(U)\otimes_{\mathscr{O}(U)}\mathscr{G}(U)$, and we hope that this defines a sheaf, which is the tensor product of two sheaves $\mathscr{F}, \mathscr{G}$. But this is not true in general. We need to modify the definition of tensor product a bit, and check that it is an invertible sheaf.
Definition 3.1. Let $\mathscr{F}, \mathscr{G}$ be two invertible sheaves of $\mathscr{O}$-module, and $\{U_i\}$ is an open cover of $X$ such that $\mathscr{F}, \mathscr{G}$ are both trivialized on $U_i$ for all $i$, then we define the tensor product of $\mathscr{F}$ and $\mathscr{G}$ as follows
$$\mathscr{F}\otimes_{\mathscr{O}}\mathscr{G}=\{(s_i)|\prod_{i}\mathscr{F}(U\cap U_i)\otimes\mathscr{G}(U\cap U_i)|s_{i_{U\cap U_i\cap U_j}}=s_{j_{U\cap U_i\cap U_j}} \}$$
We need to check
Lemma 3.2. $\mathscr{F}\otimes_\mathscr{O} \mathscr{G}$ defined above is an invertible sheaves of $\mathscr{O}$-module.
Proof. We need to check two things, first $\mathscr{F}\otimes_\mathscr{O} \mathscr{G}$ is a sheaf, and it has local generator. First, it can be seen that $\mathscr{F}\otimes_\mathscr{O} \mathscr{G}$ is a presheaf, we need to check the gluing condition. Let $V$ is an open subset of $X$, and $\{V_k\}$ is an open cover of $V$. If we are given $(s^k_i)$ in $\prod_{i}\mathscr{F}\otimes_\mathscr{O} \mathscr{G}(V_k)$ and $(s^l_i)$ in $\prod_{i}\mathscr{F}\otimes_\mathscr{O} \mathscr{G}(V_l)$ such that $s^k_{i_{V_k\cap V_l}}=s^l_{i_{V_k\cap V_l}}$. If $f_i, g_i$ are local generators for $\mathscr{F}(U_i), \mathscr{G}(U_i)$ respectively, then we can write $s^k_i=r_{ik}f_i\otimes g_i$, where $r_{ik}\in \mathscr{O}(V_k)$ , and $s^l_i=r_{il}f_i\otimes g_i$, where $r_{il}\in \mathscr{O}(V_l)$. Due to the condition that $s^k_i=s^l_i$ on the intersection $V_k\cap V_l$, we have $r_{ik_{V_k\cap V_l}}=r_{il_{V_k\cap V_l}}$, and hence, one can glue $\{r_{ik}\}$ to obtain a regular map $r_i\in \mathscr{O}(U)$. We then define $s_i=r_if_i\otimes g_i$. Then it can be seen that $s_{i_{V_k}}=s^k_i$, for all $k$. And $(s_i)\in \prod_{i}\mathscr{F}(U\cap U_i)\otimes_{\mathscr{O}(U\cap U_i)}\mathscr{G}(U\cap U_i)$. This yields such $(s_i)$ exists and unique, and hence $\mathscr{F}\otimes_\mathscr{O} \mathscr{G}$ is a sheaf. Also, one can check directly that $f_i\otimes g_i$ is the local generator for $\mathscr{F}\otimes_\mathscr{O}\mathscr{G}(U_i)$. Hence, it is an invertible sheaf. (Q.E.D)
And hence, we have defined a binary operation on the set of invertible sheaves of a curve $X$. We can easily check that for all invertible sheaves $\mathscr{F}, \mathscr{G}, \mathscr{H}$ on a curve $X$, we have $\mathscr{F}\otimes_\mathscr{O} \mathscr{F}\cong\mathscr{O}\otimes_\mathscr{O} \mathscr{F}\cong\mathscr{F}$, and $(\mathscr{F}\otimes_\mathscr{O}\mathscr{G})\otimes_\mathscr{O}\mathscr{H}\cong\mathscr{F}\otimes_\mathscr{O}(\mathscr{G}\otimes_\mathscr{O} \mathscr{H})$.
We now need to find the inverse of an invertible sheaf $\mathscr{F}$. This should be the dual of $\mathscr{F}$, and one hope it is of the form $Hom_{\mathscr{O}(U)}(\mathscr{F}(U), \mathscr{O}(U))$. But unfortunately, this is in general is not a sheaf, and we need to sheafify this presheaf.
Definition 3.3. Let $\mathscr{F}$ be an invertible sheaf, then the inverse of $\mathscr{F}$ is defined as
$$\mathscr{F}^{-1}(U)=\{(\phi_i)\in \prod_i Hom_{\mathscr{O}(U\cap U_i)}(\mathscr{F}(U\cap U_i), \mathscr{O}(U\cap U_i))|\phi_{i_{U\cap U_i\cap U_j}}=\phi_{j_{U\cap U_i\cap U_j}}\}$$
Again, similar to the proof of Lemma 3.2, we have
Lemma 3.4. Let $\mathscr{F}$ be an invertible sheaf and $\mathscr{F}^{-1}$ defined as above, then $\mathscr{F}^{-1}$ is also an invertible sheaf, and $\mathscr{F}\otimes_{\mathscr{O}}\mathscr{F}^{-1}\cong\mathscr{O}$
Proof. The first part of the statement above is proved similarly to Lemma 3.2. For the second part, if $f_i$ is the local generator of $\mathscr{F}(U_i)$, we then choose $\phi_i\in \mathscr{F}^{-1}(U_i)$ such that $\phi_i(f_i)=1$, then $\phi_i$ is the local generator for $\mathscr{F}^{-1}$. Consider the sheaf morphism $\mathscr{F}(U_i)\otimes_{\mathscr{O}(U_i)} \mathscr{F}^{-1}(U_i)\to \mathscr{O}(U_i)$ sending $r_i f_i\otimes \phi_i$ to $r_i$. It is an isomorphism of sheaves. (Q.E.D)
In short, we have proved in this section
Proposition 3.5. Denote $Inv(X)$ the group of invertible sheaves modulo isomorphism. Then $Inv(X)$ is a group, where the group operation is the tensor product of sheaves, and the identity element is $\mathscr{O}$.
4. Picard group and and invertible sheaves modulo isomorphism. If $D_1\sim D_2$ in $Div(X)$, then $\mathscr{O}[D_1]\cong \mathscr{O}[D_2]$ via the multiplication map with $f$, where $div(f)=D_1-D_2$. Hence, each $D\in Pic(X)$ defines one and only one $\mathscr{O}[D]\in Inv(X)$. This yields a well-defined map $\phi: Pic(X)\to Inv(X)$ sending $D$ to $\mathscr{O}[D]$. We will check that $\phi$ is a group homomorphism, i.e. $\phi(D_1+D_2)\cong\mathscr{O}[D_1]\otimes_\mathscr{O}\mathscr{O}[D_2]$.
For any point $p\in X$, we choose an open neighborhood $U$ of $p$ as in the proof of Proposition 2.3. i.e. $U$ does not contain zeros (except $p$) and poles of $z$, the local coordinate at $p$, together with the support of $D_1, D_2$. Then in $U$, the local generators of $\mathscr{O}[D_1]_U$ and $\mathscr{O}[D_2]_U$ are $z^{-D_1(p)}$ and $z^{-D_2(p)}$, respectively. This then yields a bilinear map from $\mathscr{O}[D_1](V)\times \mathscr{O}[D_2](V)\to \mathscr{O}[D_1+D_2](V)$, for any subset $V\subset U$, sending $(rz_1^{-D(p)}, sz^{-D_2(p)})$ to $rsz^{-D_1(p)-D_2(p)}$, where $r,s$ is regular in $\mathscr{O}(U)$. It then induce the linear map $\mathscr{O}[D_1](V)\otimes_\mathscr{O}\mathscr{O}[D_2](V)\to \mathscr{O}[D_1+D_2](V)$, sending $(z^{-D_1(p)}\otimes z^{-D_2(p)})$, i.e. it sends local generator to local generator, and hence, it is a isomorphism of sheaf. This yields $\phi$ is a group homomorphism between $Pic(X)$ and $Inv(X)$.
We assume for now any isomorphism class of $Inv(X)$ contains an element of the form $\mathscr{O}[D]$, for $D\in Div(X)$ (which we be proved in Section 5). Then
Proposition 4.1. The morphism $\phi$ defined above is an isomorphism.
Proof. We will check that $\phi$ is injective and onto group homomorphism. The part onto is trivial if we temporarily accept the result of Section 5. For the injective part, we will prove that if there is $D\in Pic(X)$ such that $\mathscr{O}[D]\cong \mathscr{O}[0]$, then $D=0$ in $Pic(X)$.
First, if $D\in Div(X)$ such that $\mathscr{O}[D]$ is isomorphic to $\mathscr{O}$, then due to Lemma 1.2, the map $\mathscr{O}[D](X)\to \mathscr{O}(X)$ is an isomorphism, and hence, there is an inverse image $f$ of $1\in \mathscr{O}(X)$ in $\mathscr{O}[D](X)$. Due to Proposition 2.1, $f$ is a local generator for all open subset $U\subset X$. On the other hand, we have $\mathscr{O}[-Div(f)](U)=\{gf|g\in \mathscr{O}(U)\}=\mathscr{O}[D](U)$. This yields $\mathscr{O}[D]=\mathscr{O}[-Div(f)]$ as sheaves.
Actually, we will prove if $D_1,D_2\in Div(X)$ and $\mathscr{O}[D_1]=\mathscr{O}[D_2]$ as sheaves, then $D_1=D_2$. If $D_1(p)> D_2(p)$ at some point $p\in X$, then by Proposition 2.3, we obtain two different local generators at neighborhood at $p$, which are $z^{-D_1(p)}, z^{-D_2(p)}$, and they are different from a multiplication of $z^{-D_1(p)+D_2)p)}$, which is not regular at $p$, a contradiction. Hence $D_1=D_2$.
From this, we can see $D=-Div(f)=Div(1/f)$, and hence $D=0$ in $Pic(X)$. (Q.E.D)
5. The proof of the Important Remark. We now complete the proof of Proposition 4.1, by construction again. If we begin with any invertible sheaf $\mathscr{F}$, we then point out some $D\in Div(X)$ such that $\mathscr{F}\cong \mathscr{O}[D]$.
Let $\{U_i\}$ be an open cover of $X$ such that over $U_i$, $\mathscr{F}_{U_i}$ can be trivialized. Let $f_i$ be a local generator for $\mathscr{F}(U_i)$. Note that we don't know what $f_i$ is, and it may not be function in general. But now, on $U_i\cap U_j$, there exists $t_{ij}\in \mathscr{O}(U_i\cap U_j)$, and $t_{ij}$ is nowhere zero on $U_i\cap U_j$. It can be seen that $t_{ij}$ satisfy the cocycle conditions.
In particular, one can see $t_{0i}\in \mathscr{O}(U_0\cap U_i)$, and it has no zero or pole there. But maybe in $U_i\setminus (U_0\cap U_i)$, it has zeros and poles. We then define $D(p):=-ord_p(t_{i0})$ for all $p\in U_i$. This is well-defined, since if $p\in U_i\cap U_j$, we have $t_{i0}=t_{ij}t_{j0}$, and $t_{ij}$ is nowhere zero in $U_i\cap U_j$, hence $ord_p(t_{i0})=ord_p(t_{j0})$. We then obtain a divisor $D\in Div(X)$, since $\{U_i\}$ cover $X$. Note that by this construction, we obtain $t_{0i}$ is the local generator for $\mathscr{O}[D](U_i)$, by Proposition 2.3.
And by Lemma 1.2, we need to check that for all open subset $U\subset X$, there exists an isomorphism $\mathscr{O}[D](U)\to \mathscr{F}(U)$. Because we are working with sheaves, it is sufficient for us to prove it for $U\subset U_i$, because we can break $U$ into $\cup_{i}(U\cap U_i)$, and each $U\cap U_i\subset U_i$, then gluing things together.
And due to Proposition 2.2, it is sufficient for us to give an isomorphism from $\mathscr{O}[D](U_i)$ to $\mathscr{F}(U_i)$. However, both are free $\mathscr{O}(U_i)$-module of rank 1, and the map sending $f_i$ to $t_{i0}$ extended linearly to $\mathscr{O}(U_i)$ gives us the desired isomorphism.
Hence, to conclude, for any invertible sheaf $\mathscr{F}$ of $\mathscr{O}$-module, there exists $D\in Div(X)$ such that $\mathscr{O}[D]\cong \mathscr{F}$. This complete the proof of Proposition 4.1.
Remark. What we can see from the first two parts is that the transition functions play an important role in construction "right" definitions. And naturally, it gives us a connection between these things and the first cohomology group as we will see later.
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