/sci/ - Science & Math

Hello, Bosch's "Algebraic Geometry and Commutative Algebra" anon here with a little update.

For those that remember my last post, I had just read through the first 200 pages (ie the first half of the book), which consist entirely of commutative algebra (with a few brief digressions into AG). It very closely resembles Atiyah/Macdonald's book, but more complete, clear and wholesome, and includes some important topics introducing homological algebra. I'd very much recommend it over A/M.

Now I've read a bit over 100 pages into the AG part of the book... And I can say it's been a bit of a ride so far. After a brief glance at Hartshorne, I can say it is basically at the same level, and covers pretty much the exact same as chapter 2 and most of chapter 3, however, again in a more complete, clear and developed way. It is very systematic, and everything seems to fall into position in just the right way. Nevertheless, given that the content is on the same level as Hartshorne, it is still quite difficult to parse through it. There are not many classical examples to rely on (in fact, the book mentions that it intentionally does not cover any classical AG), so the book still loses some motivation. However, the beginning chapters are as good as ever, where there is an informal discussion of the topics at hand, which brings some intuition to an otherwise very abstract setting.

The book enjoys a very 'top-down' approach, which introduces a lot of generality and abstractness, which results in crystal clear definitions and proofs, at the cost of losing track of your senses and intuition at times. Nevertheless, before reading this book, I had tried Eisenbud/Harris and had to leave disappointed at the unrigorous exposition, where I instead lost track because I did not have a firm grip on the concepts (perhaps I'll have a look after I'm done with Bosch). I had to leave that book after their definition of quasi-coherent ideals - completely incomprehensible despite its importance.

One of the best books that I've read. I'd be willing to say it's superior to Atiyah-Macdonald. It certainly contains almost all the material in A/M except the few digressions into Dedekind rings and the Hilbert Polynomial, which could be read independently, and on the other hand contains material on homological algebra, like derived functors and projective/injective modules. At times it felt like it was just copying A/M but the proofs and explanations are more wholesome and complete, so you're basically getting the same treatment but better. Almost at no point was a step in a proof unclear to me. There are barely any results left as exercises. Probably one of my favourite aspects are the introduction before any chapter - I've gained a lot of intuition that I previously didn't have about certain constructions and results.

There exercises are more interspersed to solidify your understanding before moving on to the next chapter. There are fewer exercises, but to be fair many of A/M exercises were algebraic geometry (at the level of schemes) in disguise, and since the book devotes its second half to AG, it is reasonable.

I have only (fully) read the first half of the book devoted to commutative algebra, but I already know the second half, devoted to scheme theory, is not going to disappoint. If you are looking to learn commutative algebra, I recommend you look into this book over Atiyah/Macdonald's.

One of the best books that I've read. I'd be willing to say it's superior to Atiyah-Macdonald. It certainly contains almost all the material in A/M except the few digressions into Dedekind rings and the Hilbert Polynomial, which could be read independently, and on the other hand contains material on homological algebra, like derived functors and projective/injective modules. At times it felt like it was just copying A/M but the proofs and explanations are more wholesome and complete, so you're basically getting the same treatment but better. Almost at no point was a step in a proof unclear to me. There are barely any results left as exercises. Probably one of my favourite aspects are the introduction before any chapter - I've gained a lot of intuition that I previously didn't have about certain constructions and results.

There exercises are more interspersed to solidify your understanding before moving on to the next chapter. There are fewer exercises, but to be fair many of A/M exercises were algebraic geometry (at the level of schemes) in disguise, and since the book devotes its second half to AG, it is reasonable.

I have only (fully) read the first half of the book devoted to commutative algebra, but I already know the second half, devoted to scheme theory, is not going to disappoint. If you are looking to learn commutative algebra, I recommend you look into this book over Atiyah/Macdonald's.