e, f, and the remainder theorem September 12, 2009Posted by Akhil Mathew in algebraic number theory, algebra, number theory, commutative algebra.
Tags: Chinese remainder theorem, Dedekind domains, ramification
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So, now to the next topic in introductory algebraic number theory: ramification. This is a measure of how primes “split.” (No, definitely wrong word there…)
e and f
Fix a Dedekind domain with quotient field ; let be a finite separable extension of , and the integral closure of in . We know that is a Dedekind domain.
(By the way, I’m now assuming that readers have been following the past few posts or so on these topics.)
Given a prime , there is a prime lying above . I hinted at the proof in the previous post, but to save time and avoid too much redundancy I’ll refer interested readers to this post.
Now, we can do a prime factorization of say . The primes contain and consequently lie above . Conversely, any prime of containing must lie above , since if is an ideal in a Dedekind domain contained in a prime ideal , then occurs in the prime factorization of (to see this, localize and work in a DVR). (more…)
Extensions of discrete valuations September 5, 2009Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
Tags: absolute values, Dedekind domains, discrete valuation rings, separable extensions
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With the school year starting, I can’t keep up with the one-post-a-day frequency anymore. Still, I want to keep plowing ahead towards class field theory.
Today’s main goal is to show that under certain conditions, we can always extend valuations to bigger fields. I’m not aiming for maximum generality here though.
Dedekind Domains and Extensions
One of the reasons Dedekind domains are so important is
Theorem 1 Let be a Dedekind domain with quotient field , a finite separable extension of , and the integral closure of in . Then is Dedekind. (more…)
Hensel’s lemma and a classification theorem September 2, 2009Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
Tags: characteristic zero, discrete valuation rings, Hensel's lemma
So, I’ll discuss the proof of a classification theorem that DVRs are often power series rings, using Hensel’s lemma.
Systems of representatives
The main result that we have today is:
Theorem 1 Suppose is of characteristic zero. Then , the power series ring in one variable, with respect to the usual discrete valuation on . (more…)
Completions of fields September 1, 2009Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
Tags: absolute values, Cauchy sequences, completions, p-adic numbers
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So again, we’re back to completions, though we’re going to go through it quickly. Except this time we have a field with an absolute value like the rationals with the usual absolute value.
Definition 1 The completion of is defined as the set of equivalence classes of Cauchy sequences: (more…)
Dedekind domains August 31, 2009Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
Tags: Dedekind domains, discrete valuation rings, Krull dimension, localization, Noetherian rings, unique factorization
Today’s (quick) topic focuses on Dedekind domains. These come up when you take the ring of integers in any finite extension of (i.e. number fields). In these, you don’t necessarily have unique factorization. But you do have something close, which makes these crucial.
DVRs II August 30, 2009Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
Tags: discrete valuation rings, Noetherian rings, PIDs, prime ideals, UFDs
Earlier I went over the definition and first properties of a discrete valuation ring. Today, it’s time to say how we can tell a ring is a DVR–it turns out to be not too bad, which is nice because the properties we need in this criterion are often easier to work with than the existence of some discrete valuation.
Today’s result is:
Theorem 1 If the domain is Noetherian, integrally closed, and has a unique nonzero prime ideal , then is a DVR. Conversely, any DVR has those properties. (more…)
Lifting idempotents à la Grothendieck August 29, 2009Posted by Akhil Mathew in algebra, algebraic geometry, commutative algebra.
Tags: completions, connectedness, idempotents, lifting idempotents, schemes
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I am going to get back shortly to discussing algebraic number theory and discrete valuation rings. But this tidbit from EGA 1 that I just learned today was too much fun to resist. Besides, it puts the material on completions in more context, so I think the digression is justified.
The theorem says we can lift “approximate idempotents” in complete rings to actual ones. In detail:
Theorem 1 Let be a ring complete with respect to the -adic filtration. Then if is idempotent (i.e. ) then there is an idempotent such that reduces to . (more…)
Discrete valuation rings and absolute values August 28, 2009Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
Tags: absolute values, discrete valuation rings, p-adic absolute value, principal ideal domains
I was initially planning on doing a post on Hensel’s lemma. Actually, I think I’ll leave that for later, after I’ve covered some more number theory (which may motivate it better).
So the goal for the next several posts is to cover some algebraic number theory, eventually leading into class field theory. At least in the near future, I intend to keep everything purely local. Thus, the appropriate place to start is to discuss discrete valuation rings rather than Dedekind domains.
Actually, it is perhaps more logical to introduce discrete valuations as a special case of absolute values, which in turn generalize the standard absolute value on . (more…)
The finite presentation trick and completions August 27, 2009Posted by Akhil Mathew in algebra, category theory, commutative algebra.
Tags: abelian categories, completions, finite presentations, flatness, Noetherian rings
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The previous post got somewhat detailed and long, so today’s will be somewhat lighter. I’ll use completions to illustrate a well-known categorical trick using finite presentations.
The finite presentation trick
Our goal here is:
Theorem 1 Let be a Noetherian ring, and an ideal. If we take all completions with respect to the -adic topology,
for any f.g. -module . (more…)
Completions of rings and modules August 25, 2009Posted by Akhil Mathew in algebra, commutative algebra.
Tags: Artin-Rees lemma, completions, exact functors, Hilbert basis theorem, Noetherian rings
So, we saw in the previous post that completion can be defined generally for abelian groups. Now, to specialize to rings and modules.
The case in which we are primarily interested comes from a ring with a descending filtration (satisfying ), which implies the are ideals; as we saw, the completion will also be a ring. Most often, there will be an ideal such that , i.e. the filtration is -adic. We have a completion functor from filtered rings to rings, sending . Given a filtered -module , there is a completion , which is also a -module; this gives a functor from filtered -modules to -modules. (more…)