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The book everyone seems to use to study Algebraic Geometry is Hartshorne's book. However, I hear a good number of people saying that this book totally misses the functorial point of view. Hence, could you please recommend a good source to learn AG using the Functor of Points approach? Thanks!!

flag	asked Jan 19 2011 at 2:16 Brian 886●2●15

No such treatment exists, although there are disconnected sets of lecture notes that give some idea of what a treatment would look like. – Harry Gindi Jan 19 2011 at 2:44

That's what I thought as well. Could you point me to those sets of lecture notes? Thanks! – Brian Jan 19 2011 at 2:59

ens.math.univ-montp2.fr/~toen/m2.html is a good one. It's might also be worth reading Neil Strickland's paper on formal groups and formal schemes, but be warned: It only covers the affine case. – Harry Gindi Jan 19 2011 at 3:20

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Another good supplement is probably Demazure-Gabriel. It doesn't use the theory of Grothendieck topologies, though. It introduces the idea of the locally ringed space associated with a scheme (as a functor of poitns) as being its "geometric realization". – Harry Gindi Jan 19 2011 at 4:17

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I should note, Brian, that I tried to do what you're doing now, and it hasn't really worked. I still basically know nothing about the geometry part of algebraic geometry. If you're really set on learning algebraic geometry and not commutative algebra + category theory, I think it might be prudent to take a more traditional swing at it. A lot of the people I've met who dislike Hartshorne have had a better time getting through EGA, which is very methodical and includes full proofs. Hartshorne is essentially worthless as a textbook if you don't do the exercises! – Harry Gindi Jan 19 2011 at 4:31

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Emerton · Answer 1 · 2011-01-19 06:59:23Z UTC

Let me second the recommendation of Mumford's Lectures on curves on an algebraic surface, which is really fantastic. Mumford's Red Book, although at a more basic level, is also very good.

In a slighty different direction: if you have succeeded in solving a large number of Hartshorne questions, then why don't you just begin reading some research papers? If you want to learn the functor-of-points view-point, choose papers which emphasize this. Ultimately, I think that this will be more productive than looking for comprehensive and self-contained texts.

fcukier · Answer 2 · 2011-01-19 06:52:13Z UTC

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One source for this point of view is the Introduction to EGA I, Springer Verlag edition (different from the IHES version). Another one is Mumford, lectures on curves on an algebraic surface.

answered Jan 19 2011 at 6:52

fcukier
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I searched on Amazon and Springer.com for "Introduction to EGA I" but I don't think they have it. Is that the correct title? Could you please tell me where I can get it? – Brian Jan 19 2011 at 13:27

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The title is 'Elements de geometrie algebrique. I.' (up to accents), so the same as the original. I believe, What fcukier meant is that the second edition (published by Springer in 1971 in the Grundl. d. math. Wiss. series, vol 166) has an introductory chapter different from (or absent of) the original IHES version. You can find precise bibliographic information on the (english) Wikipedia page of 'Elements de geometrie algebrique'. I doubt, though I don't know, that this is still in print. – quid Jan 19 2011 at 14:31

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This introduction is indeed a wonderful text, highly recommended to anyone wishing to learn about schemes. After reading it, you will probably be convinced that schemes are the most natural idea in history. – Laurent Moret-Bailly Jan 19 2011 at 16:20

Martin Brandenburg · Answer 3 · 2011-01-19 08:02:51Z UTC

Try out "Groupes algébriques" (1970) by Michel Demazure and Pierre Gabriel. In the beginnung so called $Z$-functors (which are just functors from Rings to Sets, under appropriate set-theoretic assumptions) are studied "geometrically". In particular, you can define a quasi-coherent module on it, etc.

Georges Elencwajg · Answer 4 · 2011-01-19 12:53:42Z UTC

Dear Brian, it seems that algebraic geometers who adopt your favoured approach are essentially specialists in algebraic groups.

My favourite example would be Jantzen's Representations of Algebraic groups", Academic Press 1987, in which all of Chapter 1 (18 pages) is devoted to the functor approach you require. Let me emphasize that Jantzen doesn't limit himself to affine schemes nor to group schemes. He considers completely general schemes defined as local functors admitting an open covering (in the functor sense!) consisting of affine schemes . I am sure you'll love the ingenious but natural definitions of open subfunctor, closed subfunctor, base change ... introduced in this meaty chapter: good luck!

Niels · Answer 5 · 2011-01-19 07:51:15Z UTC

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The last chapter of Eisenbud-Harris, The geometry of schemes, GTM 197, is dedicated to the functor of points.

answered Jan 19 2011 at 7:51

Niels
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Comprehensive and self-contained treatment of Algebraic Geometry using Functor of Points approach

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