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

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!!

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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 '11 at 2:44
That's what I thought as well. Could you point me to those sets of lecture notes? Thanks! –  Brian Jan 19 '11 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 '11 at 3:20
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 '11 at 4:17
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 '11 at 4:31

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.

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Thanks a lot! For some reason, I haven't thought of reading research papers. I will look into them, as well as Mumford's Lectures on curves on an algebraic surface! –  Brian Jan 19 '11 at 13:34

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.

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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 '11 at 13:27
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 '11 at 14:31
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 '11 at 16:20
This kind of introduction has also been used by Bosch in his recent textbook on algebraic geometry and commutative algebra. –  Martin Brandenburg Oct 8 '14 at 8:41

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.

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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!

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Thanks! I will look into this. You do a very good job at promoting the book :-). –  Brian Jan 19 '11 at 13:29
Dear Georges and Brian, Arithmetic geometers also use this view-point a lot. E.g. if memory serves, my first serious exposure to it was when reading Deligne--Rapoport and Katz on moduli of elliptic curves, and Mazur's Eisenstein Ideal paper (which does involve a lot of computations with group schemes (!) --- but finite flat commutative ones). Regards, Matthew –  Emerton Jan 19 '11 at 15:55
Dear Matt, you are perfectly right, of course. What I meant was that algebraic group theorists were the ones who wrote introductory books from this functorial point of view ( another nice example being Waterhouse's Introduction to affine group schemes). I am not foolish enough to pontificate about which algebraic geometers do what in their research papers... –  Georges Elencwajg Jan 19 '11 at 16:22

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

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