Equivalence of categories involving graded modules and sheaves.

Question

Let $S$ be a graded ring with $S_0=A$ a finitely generated $\mathbb{K}$-algebra and $S_1$ a finitely generated $A$-module. Let $M$ be a graded $S$-module and $\tilde{M}$ the corresponding sheaf on $Proj(S)$. For a sheaf $\mathcal{F}$ on the affine space $X=\mathbb{A}^N_{\mathbb{K}}$ we define $$ \Gamma_*(\mathcal{F})=\bigoplus_{n \in \mathbb{Z}}\Gamma(Proj(S),\mathcal{F}(n)) .$$ We define an equivalence relation $\sim$ on graded $S$-modules: we say that $M \sim M'$ if there exists an integer $k$ such that $M_{\ge k} \simeq M'_{\ge k}$, where $M_{\ge k}:=\bigoplus_{n \ge k}M_n$. If $M$ is a graded $S$-module, we say that $M$ is quasifinitely generated if it is equivalent to a finitely generated module. I have to prove that the functors $\tilde{ }$ and $\Gamma_*$ induce an equivalence of categories between the category of quasifinitely generated graded $S$-modules (modulo the equivalence $\sim$) and the category of coherent $\mathcal{O}_{Proj(S)}$-modules.

Answer 1

I think if $S_1$ doesn't generate $S$ then your statement is false. The following is Example 4.6 in "Maps between non-commutative spaces" by S. Paul Smith.

Consider the weighted polynomial ring $S = k[x,y]$, where $x$ has degree 1 and $y$ has degree 2. Let $M = S/(x)$, and consider the graded module $M(1)$, which is $M$ with the grading shifted by $1$. Note that $M(1)$ is not isomorphic to $0$ in the category of quasi-finitely generated graded $S$-modules modulo your equivalence relation.

We claim $\widetilde{M(1)} = 0$, hence the functor $N \mapsto \widetilde{N}$ is not fullly faithful. It suffices to show $$(M(1)_x)_0 = (M(1)_y)_0 = 0,$$ since $$\widetilde{M(1)}\bigr\rvert_{D_+(x)} \simeq \bigl((M(1)_x)_0\bigr)^\sim \qquad \text{and} \qquad \widetilde{M(1)}\bigr\rvert_{D_+(y)} \simeq \bigl((M(1)_y)_0\bigr)^\sim.$$ and the affine charts $D_+(x)$ and $D_+(y)$ cover $\operatorname{Proj} S$. First, $(M(1)_x)_0 = 0$ since $x \in \operatorname{Ann} M$. Next, to show $(M(1)_y)_0 = 0$, we note that $$M(1)_y = \biggl\{ \frac{f(y)}{y^n} \biggm\vert f(y) \in M(1),\ n \in \mathbf{Z}_{\ge 0} \biggr\}$$ has no non-zero degree $0$ elements, since the numerators $f(y)$ have odd degree, while the denominators $y^n$ have even degree. Thus, $(M(1)_y)_0 = 0$.

I also want to point out that similar statements relying on generation by $S_1$ (e.g. Prop. 5.12 in Hartshorne) are also false for similarly defined weighted projective spaces; see §1.5 "Pathologies" in Dolgachev's "Weighted projective spaces".

Finally, if you instead define weighted projective spaces using the stack $\operatorname{\mathbb{P}roj}S$ and look at quasi-coherent sheaves there instead, you do get an equivalence; see, e.g., Prop. 2.3 in "Mirror symmetry for weighted projective planes and their noncommutative deformations" by Auroux, Katzarkov, and Orlov.