Can someone recommend some books on commutative algebra stressing links with algebraic geometry? My concern is this. It seems to me that most of commutative algebra was formulated at least initially by algebraists, and only later were the links with geometry made more explicit. As a result, definitions which are natural to algebraists, might correspond to some complicated definitions in geometry and vice versa.

Ultimately, I would prefer a book on commutative algebra which is:
1) always reinterpreting algebraic definitions geometrically (so in some sense, written for geometers)
2) containing a lot of examples, which can be used as counterexamples to various claims, and thus exposing, rather than hiding, the subtleties of the various dictionaries between algebra and geometry
3) preferably not too big (so that it could be read entirely in a reasonable amount of time).

I have read most of Atiyah-Macdonald, and own a copy of Eisenbud's "Commutative Algebra with a view toward Algebraic Geometry". I love both books, but would like to know whether some other excellent books exist, particularly with a strong geometric bias.

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I think your assumptions are wrong (not that it is important for the issues). Arguably, one of the first books on commutative algebra was written by Zariski and Samuel with the explicit intention of codifying the algebra necessary for their work in algebraic geometry. It still is one of the deepest books in the field, though not easy to read. For example, it proves Zariski's main theorem (a very important theorem in algebraic geometry) in the strongest form, which is difficult to find elsewhere. It also deals with resolution of singularities at least for surfaces. A short, but extremely well written book on the subject is Serre's Local Algebra. Another classic is Nagata's Local Rings, again proves many theorems useful in geometry, it is short and has probably some of the best counter examples. Last, but not least is the book by Kunz, where the results are oriented towards geometry, but with a special emphasis on problems related to equations defining varieties.

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  • $\begingroup$ Thank you for all these references! I have a lot of reading to do! $\endgroup$ – Malkoun May 27 '16 at 20:54
  • $\begingroup$ I looked at the index of serre's local algebra. Could not find anything about resolution of singularities. Please give some page number or some thing like that. $\endgroup$ – user311526 May 28 '16 at 19:47
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    $\begingroup$ I did not say Local Algebra has anything to do with resolution of singularities. It is in Zariski-Samuel, of course well disguised in the theory of valuations. $\endgroup$ – Mohan May 28 '16 at 20:21

1) There is such a great book (of 120 pp.) which is Very easy to read but it ofcourse it covers less material (than Eisenbud's book) in its 120 pages. -- The book is: Miles Reid, Undergraduate Commutative Algebra. --With immediate applications to Algebraic Geometry with many examples and Pictures. Moreover, it is written for begining grad. students. 2) Afterwards, you may go back to Eisenbud's book.

3) Btw, a QUICK introduction to Algebraic Geometry (esp. for algebraic groups)
while Stating the results of Commutative Algebra WITHOUT PROOFS are covered in (i) in 20 pages in: T.A.Springer, Linear Algebraic Groups, 2nd ed. (ii) in 50 pages in: J. Humphreys, Linear algebraic Groups. Then you may consult other books of M. Reid then Eisenbud.

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  • $\begingroup$ Thank you Nazih Nahlus for the added references. Now with Mohan's references, yours, and the two that I knew about before, Atiyah-Macdonald and Eisenbud, I have a lot of fun reading ahead. $\endgroup$ – Malkoun May 29 '16 at 6:34

What about Eisenbud's Commutative algebra with a view toward algebraic geometry?

Now, I haven't read much of the thing, and I'm hardly knowledgeable about much algebraic geometry, but I enjoyed what I read and the title seems to be exactly what you're looking for.

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  • $\begingroup$ I love this book. The only thing is, I wish it was a bit shorter. It is excellent though, but it would take me a very long time to read and digest most of it. I guess I could select topics, along the way some authors suggest at the beginning of their book. So my question is whether there exist similar books to Eisenbud's book, only perhaps a bit shorter. If such a book does not exist, I will select my reading from Eisenbud's book. $\endgroup$ – Malkoun May 27 '16 at 19:23

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