TITLE: What does $\bigoplus_{n \in \mathbb{Z}} \mathcal{O}_X(n)$ represent geometrically?
QUESTION [7 upvotes]: For simplicity let's say $A$ is a noetherian ring, $S = A[x_0, \ldots, x_r]$, and $X = \operatorname{Proj} S = \mathbb{P}^r_A$.  I want to understand what, if anything, the sheaf $$\mathscr{F} := \bigoplus_{n \in \mathbb{Z}} \mathcal{O}_X(n)$$ means geometrically.
First, I think that if I put $$Y := \operatorname{Spec}(S) \setminus V(S_+) = \mathbb{A}^{n+1}_A \setminus V(S_+)$$ then I have a morphism $q : Y \to X$ (which takes prime ideals corresponding to closed points to their homogenizations) and that $\mathscr{F} = q_\ast \mathcal{O}_Y$.  Is this correct?  I'm not confident enough with these things yet to trust myself on something like that yet.  [EDIT: To be more specific, I'm having difficulty constructing the underlying function $q : Y \to X$ in a way that makes it tractable to show that $q$ is Zariski-continuous, to the point that I'm wondering whether such a map exists in general.]
Second, whether the above is correct or not, does $\mathscr{F}$ have a nice geometric meaning of some sort?

REPLY [2 votes]: The map $q$ is well defined for any positive graded ring $S$.
For any prime ideal $\mathfrak p$ of $S$, denote by $\mathfrak p^h$ the homogeneous ideal associated to $\mathfrak p$. It is also prime. For any homogeneous ideal $I$, by construction we have: 

$I\subseteq \mathfrak p^h $ if and only if $I\subseteq \mathfrak p$. 

Taking $I=S_+$, we see that $q(\mathfrak p):=\mathfrak p^h\in \mathrm{Proj}(S)$ if $\mathfrak p\notin V(S_+)$. This shows that $q$ is actually a map
$$q: \mathrm{Spec}(S) \setminus V(S_+)\to \mathrm{Proj}(S).$$
Moreover the above equivalence shows that 
$$ q^{-1}(V_+(I))=V(I) \setminus V(S_+). $$
Hence $q$ is continuous. Finally, for any $f\in S_+$ homogeneous, we have 
$$q^{-1}(D_+(f))=D(f).$$ 
This helps to show the equality of sheaves you are after.