Sheaves of $\mathscr{O}_X$-module and Sheafification

This is a very short note due to Gathmann's note on AG (version 2002, Chap VII.1). Let $(X,\mathscr{O}_X)$ be a ringed space. Recall that $\mathscr{O}_X$ is a structure sheaf of $X$, i.e. for any open set $U\subset X$, we have the ring $\mathscr{O}_X(U)$. We want to extend this notation, for example, we call $\mathscr{F}$ a sheaf of $\mathscr{O}_X$-module if for any open set $U\subset X$, we have $\mathscr{F}(U)$ is an $\mathscr{O}_X(U)$-module, and also, other axioms of sheaf, i.e. compatible with restriction map, or gluing properties also holds for $\mathscr{F}$, but we change ring (ring homomorphism) by module (module homomorphism) everywhere. This extended notation gives us a convenient way to consider working with sheaves is very similar to working with modules over commutative rings. And this leads to the very important theory, the sheaf cohomology. The following definition is essential.

Definition 1. Let $\mathscr{F}_1,\mathscr{F}_2$ be sheaves of $\mathscr{O}_X$-modules. We define the morphism $f:\mathscr{F}_1\to \mathscr{F}_2$ is the following data

(i) For any open subset $U\subset X$, the $\mathscr{O}_X(U)$-module homomorphism $f_U:\mathscr{F}_1(U)\to \mathscr{F}_2(U)$.

(ii) $f$ compatible with the restriction map, i.e. if $V\subset U$ are two open sets of $X$, and $p_{U,V}$ the restriction map from $\mathscr{F}_i(U)\to \mathscr{F}_i(V)(i=1,2)$, we have $f_V\circ p_{U,V}=p_{U,V}\circ f_U$.

Definition 2. Let $f$ be a morphism between two sheaves $\mathscr{F}_1$ and $\mathscr{F}_2$ of $\mathscr{O}_X$-module. We define the kernel of $f$, denote $\ker f$ is the following data: for any $U\subset X$ is open, $\ker f(U):=\ker (f_U: \mathscr{F}_1(U)\to \mathscr{F}_2(U))$.

A very nice properties of the kernel is

Proposition 3. Let $f$ be a morphism between two sheaves $\mathscr{F}_1$ and $\mathscr{F}_2$ of $\mathscr{O}_X$-module, then $\ker f$ is also a sheaf of $\mathscr{O}_X$-module.

Proof. It can be checked easily that $\ker f$ is the presheaf of $\mathscr{O}_X$-module, and we will check the gluing properties. Let $U=\cup_i U_i$ be an open covers of an open subset $U$ of $X$. Assume that $s_i\in \ker U_i$ such that $s_i$ and $s_j$ agree on the overlap. Because $s_i\in \mathscr{F}_1(U)$, we can glue $s_i$ to get $s$, such that $s_{U_i}=s_i$.  We now use the property (ii) of Definition 1, note that $f_U(s)_{|U_i}p_{U,U_i}(f_U(s))=f_{U_i}(p_{U,U_i}(s))=f_{U_i}(s_i)=0$. And hence, due to the gluing property of $\mathscr{F}_2$, we have $f_U(s)=0$, and $s\in \ker f$.

(Q.E.D).

Similarly for the definition of kernel, we also have the definition of cokernel.

Definition 3. Let $f$ be a morphism between two sheaves $\mathscr{F}_1$ and $\mathscr{F}_2$ of $\mathscr{O}_X$-module. We define the cokernel of $f$, denote $\text{coker }f$ is the following data: for any $U\subset X$ is open, $\text{coker } f(U):=\text{coker } (f_U: \mathscr{F}_1(U)\to \mathscr{F}_2(U)) = \mathscr{F}_2(U)/Im(\mathscr{F}_1(U))$.

Similarly, one can check directly that cokernel is the presheaf of $\mathscr{O}_X$-module. But it is not a sheaf, due to the failure of gluing properties. That leads us to the following important definition, where we will turn a presheaf to a sheaf.

Definition 4. Let $\mathscr{F}$ be a sheaf in general. Let us denote the sheafification of $\mathscr{F}$ as $\mathscr{F}^\#$, which consists of the following data
$$\mathscr{F}^\#(U)=\{\varphi=(\varphi_P)_{P\in U}\in \prod_{P\in U}\mathscr{F}_P|\forall P\in U,\exists P\in V_P\subset U\text{ open},\psi\in \mathscr{F}(V_P), \varphi_P=(V_P,\psi)\in \mathscr{F}_P\}$$
where $\mathscr{F}_P$ is the stalk at a point $P$.

First, it can be seen that for any presheaf $\mathscr{F}$, and $U\subset X$ is open, the product $\prod_{P\in U}\mathscr{F}_P$ is a sheaf. And one can see from the definition above that $\mathscr{F}^\#$ is a presheaf (It is induced from the restriction map $\prod_{P\in U}\mathscr{F}_P \to \prod_{P\in V}\mathscr{F}_V$, for any $V\subset U$ are open), and it is a subpresheaf of $\prod_{P}\mathscr{F}_P$. And also, for all $P\in X$, $\mathscr{F}^\#_P=\mathscr{F}_P$ from the definition, i.e. the sheafification preserves the stalks. And what we want is

Proposition 5. The sheafification of a presheaf $\mathscr{F}$ is a sheaf.

Proof. We have proved $\mathscr{F}^\#$ is a presheaf, and it is sufficient to prove the gluing properties. Let $U\subset X$ be any open subset, and $\{U_i\}$ is the open cover of $U$. Assume that we have $s_i\in \mathscr{F}^\#(U_i)$ such that $s_i$ and $s_j$ agree on the overlap. Because $\mathscr{F}^\#(U)$ is a subset of $\prod_{P\in U}\mathscr{F}_P$, there exists $s\in \prod_{P\in U}\mathscr{F}_P$, such that $s_{U_i}=s_i$. And we will prove that $s\in \mathscr{F}^\#(U)$, i.e. $s=(s_P)_{P\in U}$, such that each $s_P$ correspond to some $(V, \psi)$.

From the definition, take any $P\in U$, there exists $i$ such that $P\in U_i$, and  we have $s_i=(s_{i,P})_{P\in U_i}$, where $s_{i,P}$ corresponds to $(V_P,\varphi_{V_P})$, for some $V_P\subset U$ is an open neighborhood of $P$. Because $s_{i,P}=s_P$, we obtain that $s\in \mathscr{F}^\#(U)$, and $\mathscr{F}^\#$ is a sheaf.

(Q.E.D)

And also

Proposition 6. If $\mathscr{F}$ is a sheaf, then the sheafification $\mathscr{F}^\#=\mathscr{F}$.

From this, one can consider the sheafification of cokernel, tensor product, direct product (it is a sheaf itself) of two sheaves of $\mathscr{O}_X$-module. And also, the sheafification of the dual notion.