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I know intro abstract algebra and some real analysis. Is this enough to study algebraic geometry from the book of Hartshorne?

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There's no harm in trying for a day or so, but my feeling is that this isn't enough preparation. Your knowledge of analysis likely won't be the problem (outside of the appendices), but learning about varieties before learning about manifolds seems like a bad idea. You will also need to have great comfort with topology and commutative algebra -- knowing the definition of a ring and so on isn't enough. – Dylan Moreland Aug 25 '11 at 22:19
There are also far easier places to start: Reid, Shafarevich, and Kunz are all good choices. – Dylan Moreland Aug 25 '11 at 22:21
Harsthone? Do you mean Hartshorne? Also, which book of his? Do you mean his book on Algebraic Geometry? Please be a lot more specific in asking your question. Also, please use a tag besides (soft-question). If you are asking about the Algebraic Geometry book, consider tagging it as such. – Willie Wong Aug 25 '11 at 22:33
Short Answer: No. – Amitesh Datta Aug 25 '11 at 23:06
@Jesse I think that is certainly a good idea; I have posted my four comments above as an answer. – Amitesh Datta Aug 26 '11 at 7:46

I think you will need significantly more background in modern algebra. Firstly, you should probably learn the elements of point-set topology; I would recommend chapters 2-3 of Munkres' Topology: A First Course. Secondly, you should probably learn the theory of fields in some depth. In addition to the elements of Galois theory, a couple of important topics relevant to commutative algebra in the theory of fields include the theory of separable and inseparable extensions and the theory of transcendental extensions. I would recommend Algebra: A Graduate Course by Martin Isaacs.

After you have studied point-set topology and the theory of fields in some depth, I would recommend you to read An Introduction to Commutative Algebra by Atiyah and Macdonald and do most or all of the exercises in this textbook. (The exercises are important because, for one reason, they introduce the reader to affine schemes which are the basic "building blocks" of schemes.) If you are the sort of person who does not like to accept mathematical facts without proof (and, if you are, it is not bad by any means), then you should also read Commutative Algebra by Hideyuki Matsumura.

However, before you begin reading Matsumura's textbook, you should briefly learn the elements of homological algebra: from my reading of Matsumura, it seems that the most important aspects of homological algebra which you will need to know are: projective and injective modules, the Tor and Ext functors, and perhaps a very elementary knowledge of the language of category theory. The best place, I think, to learn this material is Appendix B of Matsumura's Commutative Ring Theory. Once you have finished Matsumura's textbook (Commutative Algebra), you probably have exactly the background necessary to read Hartshorne.

In pratice, I think most people do not read Matsumura's Commutative Algebra before Hartshorne's Algebraic Geometry but I could be wrong. If you are willing to accept facts in commutative algebra without proof, then it might be OK for you to defer your reading of Matsumura's textbook or to read Matsumura's textbook concurrently with Hartshorne. Alternatively, if you are interested in number theory (or not), you could read Qing Liu's Algebraic Geometry and Arithmetic Curves directly after Atiyah and Macdonald's textbook; the commutative algebra necessary to read Liu's textbook is minimal.

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I think Chapter I of Hartshorne is accessible without such a heavy load of prerequisites. Even the point set topology needed is minimal. I don't believe it is necessary to have all of Atiyah and Macdonald to study Chapter II. Homological algebra doesn't enter the picture until Chapter III. – Zhen Lin Aug 26 '11 at 7:57
Amitesh, @Dinesh: If I may voice my opinion as someone who already had a rather strong analysis background when first glancing at Hartshorne: I didn't feel that analysis helped me at all. The point set topology needed struck me as very basic (if not basically trivial) and rather too extensively explained than otherwise. Did topology just come too naturally to me? – t.b. Aug 26 '11 at 12:48
No problem with the off-topic. Mac Lane's book is very solid and the style a matter of taste. Since you like Rudin, I imagine that it should suit you well. Many people (e.g. J.P. May) say that Mac Lane's exposition of spectral sequences is still among the best you can find. Weibel is very good, concise modern, but unfortunately marred with lots of typos. But do look at Hilton-Stammbach as well, even if it may be on the more basic side. Also, I strongly encourage you to look at Cartan-Eilenberg. (to be continued in a few minutes) – t.b. Aug 26 '11 at 23:12
Well, it really starts at chapter IX but it might already be earlier (I haven't really worked through the entire text). There is an urban legend on Weil that supposedly happened when Weil, Halmos and Mac Lane were all professors in Chicago during the notorious Stone age. Weil was talking to some visitors in front of his office and said: We professors here are all good in some way: There are good mathematicians with bad taste (points down the hallway in direction of ML's office) bad mathematicians with good taste (same with H's office)... – t.b. Aug 27 '11 at 3:42
...and there are good mathematicians with good taste (no pointing)... So this should give a hint: Whenever something strikes you as rather clumsy, be wary of it. – t.b. Aug 27 '11 at 3:44

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