This is question 2.5 of Qing Liu.
I am new in algebraic geometry and really stuck on it and can't do anything to solve it.
The question:
Let $F$ be a sheaf on $X$. Let $\operatorname{Supp} F=\{x\in X:F_x\neq 0\}$. We want to show that in general, $\operatorname{Supp} F$ is not a closed subset of $X$. Let us fix a sheaf $G$ on $X$ and a closed point $x_0\in X$. Let us define a pre-sheaf $F$ by $F(U)=G(U)$ if $x_0\notin U$ and $F(U)= \{s\in G(U):s_{x_0}=0\}$ otherwise. Show that $F$ is a sheaf and that $\operatorname{Supp} F = \operatorname{Supp}G\setminus \{x_0\}$.
I don't know how to solve this question:
To show a pre-sheaf is a sheaf I need to check the "uniqueness" and "gluing local sections".
For the uniqueness: Let $U$ be an open subset of $X$ , $s\in F(U)$, if $x_0\notin U$ , then since $G$ is a sheaf, I don't see a problem for $F$ to be a sheaf.
If $s\in F(U)$ and $x_0 \in U$ and $\{U_i\}_i$ be an open covering of $U$,then there exists an $i_0$ such that $x_0\in U_{i_0}$. the image of $s$ in the stalk $F_{x_0}$ is $s_{x_0}$. $F(U_{i_0})=\{s\in G(U):s_{x_0}=0\}$ by definition. I don't know what to do now? (so sorry and I know this is an easy question…)
Best Answer
This is just definition-pushing, and you've already made a good start.
To see uniqueness, let $U\subset X$ be an open subset and $\{U_i\}$ an open cover of $U$. Let $s,t\in F(U)$ and let $s_i,t_i\in F(U_i)$ be their restrictions. Then the condition that $s_i=t_i$ in $F(U_i)$ means that $s_i=t_i\in G(U_i)$, which means that $s=t$ in $G(U)$ and as $F(U)\subset G(U)$, we have that $s=t$ in $F(U)$.
To check gluing, let $s_i$ be a collection of sections of $F(U_i)$ so that $s_i|_{U_i\cap U_j}=s_j|_{U_i\cap U_j}$ as elements of $F(U_i\cap U_j)$. Then this equality is also true in $G(U_i\cap U_j)$, and by the assumption that $G$ is a sheaf, there is a section $s\in G(U)$ so that $s|_{U_i}=s_i$. This implies that $s_{x_0}=0$ (if $x_0\in U$ - if not, we have nothing to worry about), as the maps $G(U)\to G(U_i)\to G_{x_0}$ commute, so $s\in F(U)$ as well and thus $F$ satisfies gluing.