suggestion regarding textbooks

Question

I want to buy some text books at graduate level. I have got enough money in my contingency grant, as there is a chance, i thought i should buy some good books not thinking about the cost.

The areas that i am interested in are

Commutative algebra, Algebraic geometry, Algebraic number theory, Homological algebra, Representation theory, Algebraic topology

Any thing a bit advanced is also welcome.

I do not want to buy so many books on same subject for the reason that there may be some repetitions in the content and all.

Please suggest some thing regarding this.

Answer 1

There are many questions of the type "reference request" here on this site asking for good books on various subjects, if you search efficiently. Also, as the comments have pointed out, freely available works are aplenty on the web.

Nevertheless, here is a list of various references I have personally used in my (masters level) studies:

• Algebraic Topology, by Hatcher. Freely and legally available on the web, so you can save your money on this one.
• Differential Forms in Algebraic Topology, by Bott and Tu. Rather unique in that it uses a primarily differential point of view to explain algtop (starting off with the deRham framework), but it covers a lot of ground in a way that not too many other books of that size do.
• An Introduction to Homological Algebra, by Weibel. One of the more readable algebra books I have as of yet encountered, and covers a lot (emphasis on "a lot") of stuff for your buck.
• Algebra, by Lang. It's a very thick book covering all kinds of algebra stuff. It's kind of encyclopeadic and doesn't specialize in anything in particular, but is nevertheless one of those classics some people think you shouldn't live without.
• Introduction to Commutative Algebra, by Atiyah and Macdonald. This is a surprisingly thin book, but not in any way lesser for that. It covers all the material you could want from a first book on the subject and more, and comes with plenty of exercises too.
• Representation Theory, A First Course, by Fulton and Harris. A solid book that covers the finite cases in the first 80-ish pages, that then switches to Lie groups and algebras for the remainder of the 500-600 pages. It covers a great deal of information on the latter part, and still manages to keep the finite cases well covered.
• The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, by Sagan. This focuses entirely on the finite and the discrete, but does that extremely well, covering much material not usually encountered in other works on repteo.
• Algebraic Number Theory, by Fröhlich and Taylor. I'd say this is a somewhat advanced book as far as theory goes, but highly readable.
• Introduction to Lie Algebras and Representation Theory, by Humphreys. It isn't the most up-to-date text on Lie algebras, but still very good (and very slick in that it is so thin, making it easy to carry around :) )

I might update this list by adding more references as I remember them, but for now, I hope that this is plenty enough…

Answer 2

Introduction to Commutative algebra by Atiyah & Macdoland, (as already mentioned by A.Sh) is a compact and wonderful book with a lot of exercises.

The Red Book of Varieties and Schemes by Mumford is a good start to learn about varieties and schemes. This is a relatively thin book, and you'd better reading this before reading Hartshorne.

Algebraic Geometry by Hartshorne is one of the classic, but hard one, mainly about schemes and (sheaf) cohomology.

An Introduction to Homological algebra by Rotman is quite a comprehensive book on homological algebra, including homology/cohomology/Ext/Tor stuff. This is a good alternative of Weibel.

An Introduction to Algebraic Topology by Rotman is a very readable algebraic topology book, mostly concentrating on algebraic aspects of algebraic topology. This book is really helpful when you are not used to dealing with commutative diagrams.