(Pre)Sheaves of abelian groups

Question

Let $X$ be a topological space, $A$ an abelian group, $p\in X$. Define the presheaf $i_p(A)$ to be $$i_p(A)(U):=\begin{cases} \text{A}, & \text{if } p\in U \\ 0, & \text{otherwise} \end{cases}$$ with restriction map $$\rho_{UV}=\begin{cases} \text{id}_A, & \text{if } p\in V \\0, &\text{elsewhere} \end{cases}$$ $\forall U\subset V$ open, $V\subseteq U$ open in $X$.

I showed this is actually a sheaf of abelian groups looking at the constant sheaf, which is the sheafification of $i_p$.

Now, I have to show that:

1. the stalk $i_p(A)$ at a point $q\in X$ is $A$ if $q \in \overline{\{p\}}$ closure of $\{p\}$ in $X$, and $(0)$ in all other cases;
2. Let $A_Y$ be the constant sheaf on $Y:=\overline{\{p\}}$ associated with the abelian group $A$. Let $i:Y\rightarrow X$ denote the natural inclusion. Show that there exists a canonical isomorphism of sheaves between $i_*A_Y$ and $i_p(A)$.

I am just started studying (pre)sheaves in algebraic geometry and I am not getting some points yet. I would appreciate if someone could help me.

Answer 1

$\textbf{(1) proof:}$

• If $q\in\overline{\{p\}}$ then $\exists V\subset X$ open such that $q\in V$. Note that we also have $p\in V$ otherwise $p\in X\setminus V$ which is closed, contradiction. The map $$(i_p(A))_q\rightarrow A$$ $$[(U,s)]\mapsto s$$ is well-defined: if $(U,s)\sim(V,t)$ then $\exists W$ open with $p,q\in W$ such that $$\rho_{U,W}(s)=\text{id}_A(s)=\text{id}_A(t)=\rho_{V,W}(t)\Rightarrow s=t.$$ It is injective: $(U,s)\sim(V,t)\Rightarrow U\cap V$ is open. Hence $p,q\in U\cap V$ and $\rho_{U,U\cap V}(s)=\rho_{V,U\cap V}(s)\Rightarrow (U,s)\sim (V,t)$. It is surjective: $(X,s)$ is a preimage of $s\in A$.
• If $q\notin\overline{\{p\}}$, take $(U,s),(V,t)\in (i_p(A))_q$. Define $W:U\cap V\cap X\setminus \overline{\{p\}}$, it is open and non nempty because $q\in U,V,X$. Now, since $i_p(A)(U)$ is trivial we have $$\rho_{U,W}(s)=0=\rho_{V,W}(t)\Rightarrow (U,s)\sim(V,t)$$ so $(i_p(A)(-))_q=0$.

$\textbf{(2) proof:}$ $A_Y(U)=\{f:U\rightarrow A, \text{continuous with discrete topology on }A\}$, $i_*A_Y(U)=A_Y(i^{-1}(U))=A_Y(U\cap Y)$. For any $U\subseteq X$ open, define $$i_*A_Y(U)\rightarrow i_p(A)(U)$$ $$F\mapsto \begin{cases} F(p), & \text{if }p\in U \\ 0, & \text{otherwise} \end{cases} $$ It is well-defined: easily check that it is compatible with restrictions $\Rightarrow$ morphism of presheaf $\Rightarrow$ morphism of sheaves. Take $q\in Y$ $$(i_*(A_y))_q\rightarrow(i_p(A))_q\xrightarrow{(1)}A$$ $$[(U,f)]\mapsto(U,f(p))\mapsto f(p).$$ It is injective: $(V,f),(U,g)$ such that $f(p)=g(p)$, then it agrees on the connected components of $p\in V\cap U\Rightarrow (V,f)\sim (U,g)$. It is surjective: $(X,f(x\mapsto a))$ for $a\in A$ is a preimage of $A$. Hence there is a canonical isomorphism $(i_*A_Y)_q\rightarrow(i_p(A))_q$.