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Let $X$ be a scheme and $\mathcal{A}$ be a sheaf of $\mathbb{Z}_{\ge 0}$-graded $\mathcal{O}_X$-algebras.

From the data above there is a construction which gives the "global Proj" $Proj \mathcal{ A} \to X$ which satisfies that the pullback along any affine open subset $SpecB=U \subset X$ is the (usual) Proj construction of the graded $B$-algebra $\Gamma(U,\mathcal{A})$.

I've went through several proofs of this construction over the past few months and was rather patient hoping that eventually I'll be convinced. Sadly though I'm still confused just as badly as I was on day 1. My problem is that all the ad-hoc arguments about gluing don't stick with me in the long run since I'm not entirely sure what object is being constructed and what properties are being verified. For the main example of what my problem is look at the following question

Suppose I choose an affine cover (assuming $X$ is qcqs) and construct $Proj \mathcal{A}$ via a gluing. Suppose now I choose a different cover, what garauntees me that the result will be the same? Moreover why do pullbacks behave nicely as in the above description?

Here is an attempt of mine to give a universal construction which might solve the issue:

The following limit (assuming it exists) seems natural from the point of view of gluing a space over a basis of open sets (the limit is indexed over the category of coverings with refinements as morphisms):

$$\lim_{\text{all covers}} \text{coeq} \{ \coprod Proj \Gamma(U_{ij}, \mathcal{A})\rightrightarrows \coprod Proj \Gamma(U_{j}, \mathcal{A}) \}$$

Is this limit isomorphic to the global Proj of $\mathcal{A}$? Perhaps after imposing some reasonable conditions on $X$?

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  • $\begingroup$ Just to be sure, all covers means covers by affine opens, right? If so, then this inverse limit should be just the coequaliser for the open cover consisting of all affine opens. And then something weird is about to happen if $\mathcal{A} = \mathcal{O}_X$ and when some $U_{ij}$ is not affine. $\endgroup$ – Ben Jun 17 '16 at 14:16
  • $\begingroup$ @Ben I think I want all covers to means all covers and not only affine ones. I revised the question to reflect some of the motivation and further confusion I have about this. $\endgroup$ – Saal Hardali Jun 17 '16 at 18:45
  • $\begingroup$ I probably don't understand the question (certainly I don't understand this stuff about co-sheaves). But you do know that $\operatorname{Spec} \mathscr A$ has a universal property, right? This should be given in the Stacks Project and EGA. In these two sources (at least in the second edition of EGA) the existence is probably shown using Grothendieck's representability criterion. $\endgroup$ – Hoot Jun 17 '16 at 18:48
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    $\begingroup$ I guess I don't understand. To me "by the universal property" seems like the best possible answer to the question of why something holds. $\endgroup$ – Hoot Jun 17 '16 at 18:57
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    $\begingroup$ @SaalHardali I have TeXed notes on a glueing free construction of the global spec which I could send you. However, they are written in german. $\endgroup$ – Hanno Jun 17 '16 at 20:18

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