How to show $\alpha_d : M_d \to \Gamma(X, \widetilde{M(d)})$ an isomorphism for sufficiently large $d$?

Question

Let $S$ be a (positively) graded ring and $X = \operatorname{Proj} S$. Suppose $S$ is generated by $S_1$ as an $S_0$ - algebra and suppose further that $S_1$ is a finitely generated $S_0$ - module. Thus there are $x_1,\ldots,x_r \in S_1$ such that $S = S_0[x_0,\ldots,x_r]$, hence $S$ is Noetherian.

Let $M$ be a finitely generated $S$ - module; necessarily $M$ is Noetherian since $S$ is. I want to show:

The natural map $\alpha_d : M_d \to \Gamma(X,\widetilde{M(d)})$ is an isomorphism for sufficiently large $d$.

Now I have been trying to get my head around this problem for many days but I don't even have a clear approach. The facts I know are patchy; one fact I do know is that because $S =S_0[x_0,\ldots,x_r]$ with the $x_i$'s in degree $1$, we have that $X= \operatorname{Proj} S$ is covered by finitely many $D_+(x_i)$'s.

However even though I know this I can't say why $\alpha_d$ is injective which is very frustrating: If $\alpha_d(m) = 0$, then there are $n_i$'s for which $x_i^{n_i}m = 0$. How can I conclude $m = 0$?

My question is: What are some general strategies of attacking problems like this?

Edit: Here are some extra thoughts now after the discussion with Martin. Since $M$ is Noetherian we have a filtration $$0 = M^0 \subseteq M^1 \subseteq \ldots \subseteq M^r = M$$ such that for every $0\leq i\leq r-1$, we have $M^{i+1}i/M^{i} \cong (S/p_i)(n_i)$ where $p_i's$ some homogeneous prime ideals of $S$. We can then consider the diagram

$$\require{AMScd} \begin{CD} 0 @>>> M^i @>>> M^{i+1} @>>> (S/p_i)(n_i) @>>> 0\\ @. @VVV @VVV @VVV\\ 0 @>>> \Gamma(X,\widetilde{M^{i}}) @>>> \Gamma(X,\widetilde{M^{i+1}}) @>>> \Gamma(X,\widetilde{S/(p_i)}(n_i)). \end{CD}$$

Now we can apply some snake lemma and induction over here as follows. Suppose we know for all $i$ that there is $d_i$ sufficently large for which $(S/p_i)(n_i)_{d_i} \cong \Gamma(X,\widetilde{S/(p_i)}(n_i))_{d_i}.$ Then in the inductive step we know that the left vertical map is an isomorphism for some $d_i$ sufficiently large, and the right most vertical map an isomorphism for some $d_{i+1}$ sufficiently large. Taking $e = \operatorname{max}(d_i,d_{i+1})$ and using the Snake lemma shows that the middle map is an isomorphism for $d \geq e$.

Conclusion: So as Martin says we can reduce to the case that $M = S/I$. But how can we even do this? Hartshorne claims his proof of Theorem 5.19 shows this but I am not seeing it.

Answer 1

We have reduced the question to showing $(S/\mathfrak{p})(n)_d \cong \Gamma(X,\widetilde{S/\mathfrak{p}}(n+d))$ for large enough $d$. We can assume $n=0$ since $(S/\mathfrak{p})(n)_d = (S/\mathfrak{p})_{n+d}$. We claim $\Gamma(X,\widetilde{S/\mathfrak{p}}(d)) \cong \Gamma(V(\mathfrak{p}),\widetilde{S/\mathfrak{p}}(d))$. This is a bijection by looking at the definition of $(~)^\sim$, since $$(S/\mathfrak{p}(d))_{(\mathfrak{q})} = ((S/\mathfrak{p})_{\mathfrak{q}})_d = 0$$ for all $\mathfrak{q} \notin V(\mathfrak{p})$, i.e., for all $\mathfrak{q} \not \supset \mathfrak{p}$.

So, by using the identification $\operatorname{Proj} S/\mathfrak{p} \cong V(\mathfrak{p})$, it suffices to show the following special case: Let $S$ be a graded integral domain, finitely generated by $S_1$ as an $S_0$-algebra, where $S_0$ is a finitely generated integral domain over $k$. Then, $S_d \cong \Gamma(X,\mathcal{O}_X(d))$ for all $d \gg 0$.

Letting $S' = \bigoplus_{n \ge 0} \Gamma(X,\mathcal{O}_X(n))$, and following the steps in Hartshorne's proof of Theorem 5.19, we arrive at the fact that $S'$ is a finitely generated $S$-module. Let $\{z_i\}$ be our generators. Then, as Hartshorne proves earlier, $yz_i \in S_{\ge n}$ for all $y \in S_{\ge n}$ for some $n$. In particular, if we take $d_0 = \max \deg z_i + n$, then $S'_d \subset S_d$ for all $d \ge d_0$, and so we are done.

Answer 2

Do you really want to show this by yourself? It is (part of) one of the main theorems in Serre's FAC (when $S_0$ is a field). You can find a proof in EGA II, 2.7.11. (i).