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Reference request: introduction to commutative algebra

I'm looking for a good book on commutative algebra covering most of (but not limited to) :

  • Basic Galois theory and Module algebra
  • Primary decomposition of ideals
  • Zariski topology
  • Nullstellensatz, Hauptidealsatz
  • Noether's normalization
  • Ring extensions
  • "Going up" and "Going down"

The emphasis is on the approach, as I would like a book giving a good geometric intuition of ring theory that I could use as a solid basis to start learning algebraic geometry.

All in all, do you remember a book that gave you a deeper geometric insight of commutative algebra ?

Community
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nael
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  • Hi nael. I think this covers much of the same ground as the earlier question that @Brandon found. I appreciate that you've specified some specific goals you have, and my hope is that answers new and old for the linked question can be made to give more specific details on the mentioned books — this seems to be the problem with a lot of these broad reference requests. – Dylan Moreland Jun 21 '12 at 15:48

1 Answers1

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My top 3 :

  1. Commutative Algebra: with a View Toward Algebraic Geometry, by D. Eisenbud, definitely. As Dylan said in the comments, “some will call it overly chatty but the geometry discussed there is worth everything”. To learn, nothing is too chatty, but to serve as a handbook, yes, this book might be a bit too chatty.

  2. Commutative Algebra, by Bourbaki, exhaustive, once you will be confortable, not to learn.

  3. Commutative Algebra I & II, by Zariski and Samuel, slightly old fashioned, but very pedagogic, and feature very interesting points of view, aimed at geometry.

Lierre
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  • I very much agree with the first recommendation. Some will call it overly chatty but the geometry discussed there is worth everything. Reid's Undergraduate Commutative Algebra might be a good warmup. – Dylan Moreland Jun 21 '12 at 15:18
  • @DylanMoreland — we totally agree on Eisenbud ! (I don't know Reid.) – Lierre Jun 21 '12 at 15:27
  • I've looked through Reid a bit. It would be a good "warmup" as Dylan suggested. Reid has a somewhat irreverent attitude, which I loved, and some good suggestions for the geometric connections. I think it's definitely worth a look. One might also want to look at Ernst Kunz' book "Introduction to Commutative Algebra and Algebraic Geometry," but maybe not as a first source to learn. – Chris Leary Jun 21 '12 at 17:12