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For a project, I am reading a paper that assumes that the reader knows a great deal more algebraic geometry than I do (I've just begun studying the subject). I need to understand what $\operatorname{Proj}$ is, at least intuitively, so that I can go on with the paper, or at least take a look at some references and understand more about $\operatorname{Proj}$ without having to read a hundred pages of algebraic geometry first. Is there anybody that could help me, please?

Daniel Robert-Nicoud
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1 Answers1

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$Proj$ is to $Spec$ as projective varieties are to affine varieties. To elaborate on this, just as $Spec$ takes a ring and gives you an affine scheme by putting a structure sheaf on the space of prime ideals with the Zariski topology, $Proj$ takes a graded ring and gives you a (general) scheme by putting a structure sheaf on the space of homogeneous prime ideals with the Zariski topology. I'm leaving out some details, but you said you wanted an "intuitive" understanding.

If you wanted a reference that could show you these constructions without requiring a huge amount of algebraic geometry background (in the sense that if you know commutative algebra, you can freely read the book and stop to look something up if it's unfamiliar to you), then I would recommend The Geometry of Schemes by Eisenbud and Harris.

Daniel Robert-Nicoud
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Nick
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  • Thank you very much. I found the book online on the site of the university of Harvard, and took the freedom to add the link to your answer. – Daniel Robert-Nicoud Feb 21 '14 at 22:23
  • Ah, thanks for that, I was off in another tab looking it up to add it and when I came back it was already there. By the time you're through with the first chapter of the book you should have what you need, but if you still need more specifics then skip the second chapter and pick up from the third chapter, which is essentially entirely about $Proj$. – Nick Feb 21 '14 at 22:24
  • The link to isites.harvard.edu is broken, but the book can also be found at doi:10.1007/b97680. – The Amplitwist Jun 20 '22 at 14:17