I'm trying to do Hartshorne's exercises on local cohomology at the moment and seem to be stuck in Exercise III 2.5. The problem goes as follows:
$X$ is supposed to be a Zariski space (i.e a Noetherian and sober topological space) and $P\in X$ a closed point. Let $X_P$ denote the subset of $X$ consisting of all points specializing to $P$, and give it the induced topology. Let $j:X_P\rightarrow X$ be the inclusion map and write $\mathcal F_P=j^{-1}\mathcal F$ for any sheaf (of abelian groups) $\mathcal F$ on $X$. Then for all $i\in\mathbb Z$ and sheaves $\mathcal F$ we have \begin{equation} H_P^i(X,\mathcal F) = H^i_P(X_P,\mathcal F_P) \end{equation}
I succeeded in showing that the identity holds for $i=0$, so to prove the claim it would suffice to know that the $H^i_P(X_P,(~\cdot~)_P)$ define a universal $\delta$-functor. I tried to do this with the standard procedure by showing effaceability of these functors for $i>0$. So first I tried to show that pulling back along $j$ preserves flsaqueness, but meanwhile I'm quite convinced that this isn't true. I do not see why injectives would go to injectives, either. Then I thought maybe for special sheaves like sheaves of discontinuous sections flasqueness is preserved, but, as before, it didn't lead anywhere. Does anyone have some good advice?
Thanks!
Here is how I would prove this:
Let $X$ be a Zariski space. Let $ p \in X$ be a closed point and let $X_{p} \subset X$ be the set of all points $ q \in X$ such that $p \in (q)^{-}$. We observe that $X_{p}$ has the induced topology. Now let $j: X_{p}\hookrightarrow X$ be the inclusion map. For any sheaf $\mathcal F$, let $\mathcal F_{p}=j^{*}\mathcal F$. We then state our claim as: $\Gamma_{p}(X, \mathcal F)=\Gamma_{p}(X, \mathcal F_{p})$. Any open set $U$ containing $p$ also contains $X_{p}$ so gluing sheaves will not affect $\Gamma(X_{p}, \mathcal F_{p})= \lim\underset {\rightarrow p \in U} \Gamma (U, \mathcal F)$. Taking $\pi \in \Gamma_{p} (X, \mathcal F)$, we get a section $\pi^{\prime} \in \Gamma_{p}(X_{p}, \mathcal F_{p})$. We may represent $\pi^{\prime}$ by $\pi \in \Gamma(U, \mathcal F)$. Shrink $U$ such that we may assume $\mathrm{Supp}(\pi)=p$. Glue $\pi$ and $0\in \Gamma (X/p, \mathcal F)$ to obtain a global section. Then there exists a bijection $\Gamma_{p}(X, \mathcal F_{p})\leftrightarrow \Gamma_{p}(X_{p}, \mathcal F_{p})$.
Finally, if $0 \rightarrow \mathcal F \rightarrow I_{0} \rightarrow I_{1} \rightarrow \cdots$ is a flasque resolution of $\mathcal F$, then $0 \rightarrow \mathcal F_{p} \rightarrow I_{0,p} \rightarrow I_{1,p} \rightarrow \cdots$ is an injective resolution of $\mathcal F_{p}$. By repeating this argument we obtain $\Gamma_{p}(X, I_{i}) \cong \Gamma_{p}(X_{p}, I_{i,p})$ and we have that $H^i_p(X, \mathcal F)=H^{i}_{p}(X_{p}, \mathcal F_{p})$.
I hope this helps!