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Somewhat related to this question, but almost infinitely more basic.

A Confession

I am, should classification prove essential, a differential geometer and a topologist by inclination and by training: as an undergraduate I shunned any ring that wasn't $\mathbb{Z}_n$ or a ring of differential operators and held close the differentiable and the non-singular. It did not seem to matter then that these exotic 'schemes' and their exciting projective morphisms were beyond me, and to a certain extent it does not seem to matter now; but increasingly my old uni friends, fellow MOers (and, hey, even math.stack exchangers) are talking about nothing else but schemes.

In recent months (after a frighteningly eye-opening MO question) I have found myself becoming more amenable to rings, and am less daunted by my paucity of understanding than previously. In spite of this, though, I remain entirely in the dark about schemes.

Where I Sit

I am not a complete novice. I completed an undergraduate course in algebraic geometry: illuminating, interesting, but all classical beyond belief. I have read and re-read the wikipedia page on schemes several times- taking in all of the neccessary components: the spectrum of a ring, a locally ringed space et al, but have no idea how these fit together to make the objects I fiddled with over a semester two years ago.

I have made numerous guesses about generalised nulstellensatze and structure sheaves, but to explain any would probably be to complicate matters further unneccessarily. I am aware there are probably brilliant texts that do exactly what I am asking for, but I am not currently affiliated to a university and my current library would require ordering in, which for the sort of toe-dipping excercise I intend here would be overkill. So I ask:

Can anyone provide me with a canonical example of a scheme, pointing along the way the topology and the spectra associated to each open set. Perhaps deeper, if it pleases: what I am looking for is a sort of 'scheme jargon safari'.

I am aware this is silly, and perhaps asking for a verbatim quotation of page 2 of any decent algebraic geometry text, but I would be ever so grateful. Can anyone help?

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    It is not really clear to me what kind of details you want. What part of the construction and its relation to the classical theory, exactly, don't you understand?2010-08-05
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    I literally have no idea how it all fits together. What I am asking is as basic as: "given a classical variety, what are the open sets and how does one associate the spectrum of a ring to these open sets, and how does this characterize the variety?"2010-08-05
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    I'm the poster of the question you linked to. I've found Ravi Vakil's notes (http://math.stanford.edu/~vakil/0708-216/) supplemented by Eisenbud and Harris's Geometry of Schemes to be a very good combination for learning about this stuff.2010-08-06

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