I've grown increasingly unhappy with the explanation given in introductory accounts of algebraic geometry for why we only consider the subring $S_{(f)}$ of degree $0$ elements in the localisation of a graded ring.

I am aware of the intuitive explanation in terms of rational functions not being well defined on projective space unless they are ratios of polynomials in the same degree. But this explanation seems rather ad hoc. I would prefer to see where this arises from universal properties.

For the specific example I had in mind, let $(X, \mathcal{O}_{X})$ be a locally ringed space (or just noetherian scheme or even just variety if you want) with locally free sheaf $\mathcal{L}$. Then we can form the graded ring $$ \Gamma_{*}(X, \mathcal{L}) = \bigoplus_{d \geq 0} \Gamma(X, \mathcal{L}^{\otimes d}). $$ Further, if $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then we can form the graded $\Gamma_{*}(X, \mathcal{L})$-module, $$ \Gamma_{*}(X, \mathcal{L}, \mathcal{F}) = \bigoplus_{d \in \mathbb{Z}} \Gamma(X, \mathcal{F} \otimes \mathcal{L}^{\otimes d}). $$ Under modest finiteness assumptions (say a noetherian scheme) one obtains an isomorphism for any $f \in \Gamma(X, \mathcal{L})$, $$ \beta: \Gamma_{*}(X, \mathcal{L}, \mathcal{F})_{(f)} \stackrel{\simeq} {\longrightarrow} \Gamma(X_{f}, \mathcal{F}). $$ This is a fundamental result for studying quasicoherent sheaves on projective schemes, and can be found, for example in Hartshorne II.5.14.

The proof is easy enough to follow, but it seems like restricting it to degree zero just obscures any kind of universal property argument. I would like to define $\beta$ as the map arising from the universal property of localisation.

To make precise what I had in mind, let $X_{f}$ be the non-vanishing set of the section $f$ and let $i: X_{f} \hookrightarrow X$ be the open inclusion. Then from the adjunction $i^{*} \dashv i_{*}$ one obtains a graded morphism of graded rings, $$ \rho: \Gamma_{*}(X, \mathcal{L}, \mathcal{F}) \longrightarrow \Gamma_{*}(X_{f}, i^{*}\mathcal{L}, i^{*}\mathcal{F}) $$ Then I would like to argue that this map factors through the localisation map, $$ \Gamma_{*}(X, \mathcal{L}, \mathcal{F}) \longrightarrow \Gamma_{*}(X, \mathcal{L}, \mathcal{F})_{f}, $$ where we used the universal property for localisation in the category of (not necessarily graded) modules.

But why do we just make an ad hoc restriction of that factorisation down to degree $0$? Can every factorisation arising in that way in the category of graded rings be uniquely obtained by its restriction to the degree $0$ component? Is there some other universal property to use here? Any insight would be appreciated.

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    $\begingroup$ This is probably not the answer you're looking for, but in my mind giving a $\mathbb{Z}$-grading to a ring is equivalent to giving a $\mathbb{G}_m$-action and taking Proj of it amounts to taking a suitably 'nice' quotient. So the map $\operatorname{Spec}(S_f) \rightarrow \operatorname{Spec}((S_f)_0)$ is also really just the quotient map. $(S_f)_0$ are your $\mathbb{G}_m$-invariant functions. I hope this makes it look less arbitrary! $\endgroup$ – loch Jun 7 '19 at 22:49
  • $\begingroup$ @loch this may actually be exactly the answer I am looking for. Is there some resource that treats this point of view? $\endgroup$ – Luke Jun 10 '19 at 14:05
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    $\begingroup$ See www-fourier.ujf-grenoble.fr/~mbrion/notes_luminy.pdf, at least if you don't mind restricting to $\mathbb{C}$ - Example 1.7 explains the equivalence between a $\mathbb{Z}$-grading and a $\mathbb{C}^*$-action; The discussion on quotients of affine varieties for what I meant when I said $\operatorname{Spec}(S_f) \rightarrow \operatorname{Spec}((S_f)_0)$ is a quotient map, to see that functions of grading $0$ correspond to $\mathbb{C}^*$ invariant function should just be a direct verification. $\endgroup$ – loch Jun 10 '19 at 16:18

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