I am trying to understand the following Lemma ((a), Lemma 5.3, page 112 Robin Hartshorne), but some parts are causing me difficulties.
Lemma 5.3) Let $X=\operatorname{Spec} A$ be an affine scheme, let $f\in A$, let $D(f) \subset X$ be the corresponding open set, and let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.
(a) If $s \in \Gamma(X,\mathcal{F})$ is a global section of $\mathcal{F}$ whose restriction to $D(f)$ is $0$, then for some $n>0$, $f^{n}s=0$.
(1) Before doing the proof, the author refers to some facts. First, he claims : $\mathcal{F}|_{D(g)}=(M \otimes_B A_{g})^{\sim} $. I try to check his claim via the following facts.
Proposition 2.3) Any moprhism of a locally ringed space from
$\operatorname{Spec} B$ to $\operatorname{Spec} A $ is induced by a homomorphism of rings $A \to B$Proposition 5.2) Let A be a ring and $X=\operatorname{Spec} A$, Also let $A \to B$
be a ring homomorphism, and let $f:\operatorname{Spec} B \to \operatorname{Spec} A $ be
corresponding morphism of spectra. Then, for any $A$-module $M$,
$f^{*}(\overset{\sim}{M})\cong (M \otimes_B A_{g})^{\sim}$
First, the author say an inculsion $D(g) \hookrightarrow V:=\operatorname{Spec} B $ corresponds to a ring homomorphism, $B \to A_{g}$. (proposition 2.3) Thus, I think that a locally ringed space(really?) $D(g)\overset{?}{=}\operatorname{Spec}~ A_{g} \to \operatorname{Spec} B $ is induced the before-mentioned ring homomorphism. And, then I myself induce the claim as :
$$ \mathcal{F}|_{D(g)} = \overset{\sim}{M} ~{\cong}~ \overset{\sim}{M} \otimes _{\mathcal{O}_X} \mathcal{O}_{X} = \overset{\sim}{M} \otimes _{i^{-1}{\mathcal{O}_X}} \mathcal{O}_{X} = i^{*}(M) \overset{Prop ~5.2}{\cong} (M \otimes_B A_{g})^{\sim}$$
Is this the right proof? To begin with, I believe(!) to hold $D(g)=\operatorname{Spec} A_{g}$ because this is the right way in order to apply Proposition 2.3) But I am not sure whether this actually holds.
(2) [Within the Textbook] : … Hence, $\mathcal{F}|_{D(g)}=(M \otimes_B A_{g})^{\sim} $. Thus we have shown that if $\mathcal{F}$ is quasicoherent on $X$, then $X$ can be covered by open sets of the form $D(g_{i})$ where for each $i$, $\mathcal{F}|_{D(g_{i})} \cong \overset{\sim}{M}_{i}$ for some module $M_i$ over the ring $A_{g_{i}}$. Since X is quasicompact, a finite number of these open sets will do.
I do not understand why $X$ is quasi-compact. It is true that $X$ can be covered by open sets of the form $D(g_{i})$, but not covered by finite open sets.
(3) [Proof of the Proposition (a)] The author claims that $\mathcal{F}|_{D(fg_{i})}= (M_i)^{\sim}_{~f}$ for some module $M_i$ over the ring $A_{g_{i}}$ while verifying the proposition (a), But I am not sure why it holds. He refers that it holds because of the following proposition.
Propostion 5.1) For any $f \in A$, the $A_{f}$-module,
$\overset{\sim}{M}(D(f))$ is isomorphic to the localization module
$M_{f}$.
It is true that $ \mathcal{F}|_{D(g_{i})} = \overset{\sim}{M}_{i}$ because $\mathcal{F}$ is a quasi-coherent sheaf. But I cannot reach to the way how to apply the above proposition.
Best Answer
Most of your issues are covered via definitions and exercises earlier in the text.