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The definition given of the twisting sheaf on $\textrm{Proj}S$ for some graded ring $S$ is $\mathcal O_S(n) = \widetilde{S(n)}$. I'm having trouble seeing what this sheaf is though, I'm not even totally sure what the module $S(n)$ is (I think it's the same set as $S$ but the grading is such that degree $i$ elements in $S(n)$ are degree $n+i$ elements in $S$?). I guess it's not helped by the construction of the sheaf associated to a module being not particularly easy to visualise.

For example of one of the things I'm having trouble understanding I've seen that $s \in S_d$ can be naturally associated to a global section of $\mathcal O_S(d)$ but I can't explicitly work out what this section should be.

Perhaps this question is too general/vague so if it is just let me know and I'll try and add in some more specific questions.

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  • $\begingroup$ Before you try to understand this in the $\operatorname{proj}$ case, do you understand how to think about the sheaf associated to a module in the affine case? For example, do you understand what the sections are over the basic opens $D(f)$? Once you get this, you can generalize to your case and the $D_+(f)$ basic opens (as Hartshorne denotes them). $\endgroup$ Jul 3, 2018 at 15:16
  • $\begingroup$ Yeah I think so. In the affine case I just think about the sheaf as the module itself, similar to the structure sheaf. I guess I am less confident in my understanding about $\textrm{Proj}$ itself, having not done much work at all on graded rings. So while I know what the notation $S_{(p)}$ means I have little intuition for it. $\endgroup$
    – Fromage
    Jul 3, 2018 at 15:37

1 Answer 1

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First, let's fix some notation to make our lives easier. Let $S_\bullet$ be a $\mathbb{Z}^{\geq 0}$-graded ring. We construct a scheme $\operatorname{Proj}(S_\bullet)$. It has a covering by affine open patches of the form $\operatorname{Spec}((S_{\bullet})_f)_0)$. In words, these affine open patches are the spectrum of the degree $0$ part of the localization of the graded ring at some homogeneous element $f$ of positive degree. These form a base for the topology on $\operatorname{Proj}(S_\bullet)$, and we can identify $\operatorname{Spec}((S_{\bullet})_f)_0)$ with $D(f)=\{p:f(p)\neq 0\}$. To define a sheaf, we only need to define a sheaf on the base, so we need to declare what the sections of our sheaf are on the $D(f)$ basic opens. Now let $M_\bullet$ be a $\mathbb{Z}$-graded $S$-module. In your case, we just take $M_\bullet=S_\bullet$. As you have in your post, we can define the graded module $M(n)_{\bullet}$ by $M(n)_{m}=M_{n+m}$. Now, we are ready to define our sheaf. We let $\Gamma(D(f),\widetilde{M(n)_{\bullet}})=((M_{\bullet})_f)_n$.

The important thing to understand here is that this is really the only definition that could possibly make sense! Indeed, it's precisely the definition that generalizes what the sheaf associated to a module is in the affine case, as we just take our sections over the basic affine opens to be the degree $0$ part of the localization of our given module $M(n)$.

Now, with these generalities in mind, how do we think about global sections of $\mathcal{O}_S(d)$? A global section is given by the data of local sections on an open covering that agree on overlaps. According to our definition, we can just think of these local sections as being of the form $\frac{s}{f}$ where $s\in S_{\bullet}$ is homogeneous and $f$ is some homogeneous element of positive degree, such that the difference in the degrees of $s$ and $f$ is $d$. But if these local sections have to agree on all the overlaps, then we can't have any denominators (you should work this out)! So we can precisely identify global sections with homogeneous degree $d$ elements of $S_{\bullet}$.

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  • $\begingroup$ Thanks for this answer it has really helped clear up my understanding. The explicit construction of the sheaf on the basic open sets seems much easier for me now, I think I was making it out to be too complicated in my head. However I do have one question: you say at the end we can precisely identify global sections of $\mathcal O_S(d)$ with homog degree $d$ elements of $S$. Do you mean a bijective correspondence here or do you mean something like a global section is given by a tuple $(s_f)_{f \in S_+}$ where these $s_f \in S$ are related 'somehow'? $\endgroup$
    – Fromage
    Jul 4, 2018 at 11:00
  • $\begingroup$ If it's the former then why doesn't this contradict Caution 5.13.1 in Hartshorne? (I have a feeling that the global section on $\mathcal O_S(d)$ include $S_d$ but sometimes contain other things as well.) $\endgroup$
    – Fromage
    Jul 4, 2018 at 11:00
  • $\begingroup$ Good point. I usually work over a field where the identification is precise. If you want to know all the gory details, take a look at chapter 15.4 in these notes: math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf $\endgroup$ Jul 4, 2018 at 14:53
  • $\begingroup$ Also exercise 15.2.A in those notes is helpful. $\endgroup$ Jul 4, 2018 at 14:56

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