Commutative algebra with a view toward algebraic _number theory_

Question

Someone asked me this today, and I don't know what the standard answer is:

Is there an analogue of David Eisenbud's rather amazing Commutative Algebra With a View Toward Algebraic Geometry but with a view toward algebraic number theory? Ideally, with the starting graduate student in mind and with a modern slant...

Answer 1

I concur that Neukirch is a good candidate, so instead of starting with a new recommendation (I'll come back to that later), let me instead disagree with Felipe Voloch's contention that algebraic number theory is all about rings of dimension one (though certainly he had a narrower scope of algebraic number theory in mind than I'm about to describe). So a quick run-down of the fundamental, and reasonably beginner grad-level, commutative algebra I've run into doing algebraic number theory, with the caveat that I've never been very good at figuring out where commutative algebra ends and some of these other things begin (in particular, commutativity tends to fade away somewhat silently):

1. Basic stuff: As mentioned above. Dedekind rings, local rings, valuation theory, integral closures, PIDs/UFDs, etc.

2. Arithmetic Geometry: Okay, okay, this one's cheating given the context of the question. But still, you can't get too far in algebraic number theory before you run into an elliptic curve, and then you'll want to know something about its function field, and so on.

3. Homological Algebra: Free and projective resolutions of groups, most poignantly with the goal of getting to Galois cohomology, which is a natural language for much of algebraic number theory. In particular, there's the cohomological version of class field theory, Cornell and Rosen's treatise on getting much of algebraic number theory cohomologically, Tate-Shafarevich groups, local-global obstructions, etc.

4. Topological Rings/Fields: e.g., rings of adeles. More generally, direct/inverse limit constructions, especially to get cohomology of profinite groups via limits.

5. Fancier Stuff: Of which there is probably no end. But I'll just mention that, e.g., Wiles's proof of FLT uses universal deformation rings, complete intersection rings, Gorenstein rings, etc. (Though some of this was subsequently tidied up a little.)

My recommendation would be to start with Neukirch's Algebraic Number Theory for roughly the first bullet point, and as the follow-up book, to go to Manin and Panchishkin's Introduction to Modern Number Theory for basically everything else in the list (with a hat tip to Lorenzini's An Introduction to Arithmetic Geometry as mentioned in the comments). For books that then make heavy use of this material, there's Neukirch et al's follow-up book Cohomology of Number Fields, and Georges Gras's Class Field Theory.

Answer 2

• Neukirch was already mentioned a couple of times.
• Atiyah-MacDonald: Introduction to Commutative Algebra
• P. Samuel: Algebraic Theory of Numbers (Has a fair amount of commutative algebra.)
• Serre: Local Fields
• Cassels-Frolich: Algebraic Number Theory (this is more advanced; not for a beginners)

Answer 3

According to Mariano's request, I turn my comment into an answer:

I think Neukirch's book "Algebraic Number Theory" might be a good reference. The first part "look reasonably abstract" to be thought of as "commutative algebra" but it concentrates on topics (dimension $1$, in particular) that point towards arithmetic applications.