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I'm a physicist and so never had a lecture on algebraic geometry. What I really try to find out for some time now (and it's funny how I don't seem to be able to find it out) is if the modern theory, since the 60's say, is concerned with new objects or if, in the end, they are concerned with objects which where already known before. And here I count an object which comes from gluing together other reasonable objects as not essentially new - I expect all the small parts (this and that ring or module) must have been known before. This question really goes for things with names like K-theory too.

So are schemes and sheaves compilations of things one had before, are they more or less "just" tool for the classification of weird but not totally far fetched spaces (the dimensions seems to be relatively low at least), or are these really new ideas? Is it that people try to understand all the spaces as well as spaces over spaces and so on using new tools, or is it about essentially new things?

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    You consider high-dimensional spaces far-fetched? $\mathbb{R}^n,$ for instance, was studied quite a while before the 60s.2012-08-13
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    What do you mean by "the dimensions seem to be relatively low"? Schemes may be very high-dimensional, in a certain sense. And on a different note: are you comfortable with say "old-fashioned" affine and projective varieties over an algebraically closed field like $\mathbb{C}$? To gain respect and admiration for the concept of a scheme it was absolutely necessary for me to know about varieties in affine or projective space and their defects, some of which were nicely resolved using the concept of a scheme, in establishing an "algebraic geometry" which "truly" deserves the "geometry" in it.2012-08-13
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    I count all spaces which would have been known to Klein or Hilbert as old ones here. The question is if the modern approaches are to classify newer spaces, or get a hold of these old ones and their combinations. "Are there some object, which I'd not know from physics, other than any algebraic structre (some modules) glued on top of some reasonable/imaginable topological space?". 'the dimensions seem to be relatively low' = fixed counable cardinality. These things which mathematicans surely have come up before Grothendieck.2012-08-13
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    Even spaces with uncountable dimension are long known, for example the vector space of all functions $f:\mathbb R\to\mathbb R$.2012-08-13
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    But, OK, if you're fine with up to countable-dimensional spaces, Nick, I don't think you'll find anything to object to on that note in modern algebraic geometry.2012-08-13
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    @celtschk: I didn't intend to imply that this isn't so. Not sure if this is directly related to the question anyway.2012-08-13
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    @KevinCarlson: Okay, then is it possible to conceptualize what the trick is in the modern formulation? Where lies the main difficulty? I seem not to be able to understand the idea of nesting of topological spaces, and so this seems to me what's going on.2012-08-13
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    @NickKidman: Could you please tell us what your background in AG is? Any books/papers you tried reading on the subject? Also what makes you think that "the idea of nesting topological spaces...seems to me what is going on"? I really don't see why this should be a major point in AG. Maybe because I don't get what you exactly mean.2012-08-13
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    @NilsMatthes: I don't have any relevant background, only some ideas through books like "Geometry, Topology and Physics" by Nakahara. In relations with schemes and sheafs, I read about "keeping track of topological data", this is where I got the vague idea from.2012-08-13
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    @NickKidman: I recommend reading some of the online lecture notes of Ravi Vakil on algebraic geometry. They are here http://math.stanford.edu/~vakil/216blog/ Especially the third chapter on sheaves and the fourth chapter on affine schemes will be helpful to get a rough idea of what schemes and sheaves are about in AG. Of course it's better to study affine and projective varieties over an alg. closed field first; this is where a lot of motivation for algebraic geometry comes from. But if you only want an overview on schemes, the mentioned lecture notes should be a good starting point.2012-08-13

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