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e, f, and the remainder theorem September 12, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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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 {A} with quotient field {K}; let {L} be a finite separable extension of {K}, and {B} the integral closure of {A} in {L}. We know that {B} 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 {\mathfrak{p} \subset A}, there is a prime {\mathfrak{P} \subset B} lying above {\mathfrak{p}}. 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 {\mathfrak{p}B \subset B,} say {\mathfrak{p}B = \mathfrak{P}_1^{e_1} \dots \mathfrak{P}_g^{e_g}}. The primes {\mathfrak{P}_i} contain {\mathfrak{p}B} and consequently lie above {\mathfrak{p}}. Conversely, any prime of {B} containing {\mathfrak{p}B} must lie above {\mathfrak{p}}, since if {I} is an ideal in a Dedekind domain contained in a prime ideal {P}, then {P} occurs in the prime factorization of {I} (to see this, localize and work in a DVR). (more…)

Extensions of discrete valuations September 5, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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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 {A} be a Dedekind domain with quotient field {K}, {L} a finite separable extension of {K}, and {B} the integral closure of {A} in {L}. Then {B} is Dedekind. (more…)

Hensel’s lemma and a classification theorem September 2, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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So, I’ll discuss the proof of a classification theorem that DVRs are often power series rings, using Hensel’s lemma. 

Systems of representatives  

Let {R} be a complete DVR with maximal ideal {\mathfrak{m}} and quotient field {F}. We let {k:=R/\mathfrak{m}}; this is the residue field and is, e.g., the integers mod {p} for the {p}-adic integers (I will discuss this more later).

The main result that we have today is:

Theorem 1 Suppose {k} is of characteristic zero. Then {R \simeq k[[X]]}, the power series ring in one variable, with respect to the usual discrete valuation on {k[[X]]}. (more…)

Completions of fields September 1, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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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 {F} with an absolute value {\left \lvert . \right \rvert} like the rationals with the usual absolute value.

 Completions   

Definition 1 The completion {\hat{F}} of {F} is defined as the set of equivalence classes of Cauchy sequences:  (more…)

Dedekind domains August 31, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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Today’s (quick) topic focuses on Dedekind domains. These come up when you take the ring of integers in any finite extension of {\mathbb{Q}} (i.e. number fields). In these, you don’t necessarily have unique factorization. But you do have something close, which makes these crucial. 

Definition 1 A Dedekind domain is a Noetherian integral domain {A} that is integrally closed, and of Krull dimension one—that is, each nonzero prime ideal is maximal.   (more…)

DVRs II August 30, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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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 {R} is Noetherian, integrally closed, and has a unique nonzero prime ideal {\mathfrak{m}}, then {R} is a DVR. Conversely, any DVR has those properties. (more…)

Lifting idempotents à la Grothendieck August 29, 2009

Posted by Akhil Mathew in algebra, algebraic geometry, commutative algebra.
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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. 

Lifting Idempotents

The theorem says we can lift “approximate idempotents” in complete rings to actual ones. In detail: 

Theorem 1 Let {A} be a ring complete with respect to the {I}-adic filtration. Then if {\bar{e} \in A/I} is idempotent (i.e. {\bar{e}^2=\bar{e}}) then there is an idempotent { e \in A} such that {e} reduces to {\bar{e}}  (more…)

Discrete valuation rings and absolute values August 28, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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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. 

Absolute Values  

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 {\mathbb{R}}(more…)

The finite presentation trick and completions August 27, 2009

Posted by Akhil Mathew in algebra, category theory, commutative algebra.
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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 {A} be a Noetherian ring, and {I} an ideal. If we take all completions with respect to the {I}-adic topology,      

 

\displaystyle \hat{M} = \hat{A} \otimes_A M

for any f.g. {A}-module {M}.   (more…)

Completions of rings and modules August 25, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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So, we saw in the previous post that completion can be defined generally for abelian groups. Now, to specialize to rings and modules.

 Rings

 The case in which we are primarily interested comes from a ring {A} with a descending filtration (satisfying {A_0 =A}), which implies the {A_i} are ideals; as we saw, the completion will also be a ring. Most often, there will be an ideal {I} such that {A_i = I^i}, i.e. the filtration is {I}-adic. We have a completion functor from filtered rings to rings, sending {A \rightarrow \hat{A}}. Given a filtered {A}-module {M}, there is a completion {\hat{M}}, which is also a {\hat{A}}-module; this gives a functor from filtered {A}-modules to {\hat{A}}-modules. (more…)

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