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Dedekind domains August 31, 2009

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

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.  

A DVR is a Dedekind domain, and the localization of a Dedekind domain at a nonzero prime is a DVR by this. Another example (Serre) is to take a nonsingular affine variety {V} of dimension 1 and consider the ring of globally regular functions {k[V]}; the localizations at closed points are DVRs, so the ring is a Dedekind domain.

Now assume {A} is Dedekind.

A f.g. {A}-submodule of the quotient field {F} is called a fractional ideal; by multiplying by some element of {A}, we can always pull a fractional ideal into {A}, when it becomes an ordinary ideal. The sum and product of two fractional ideals is a fractional ideal. 

Theorem 2 (Invertibility) If {I} is a nonzero fractional ideal and { I^{-1} := \{ x \in F: xI \subset A \}}, then {I^{-1}} is a fractional ideal and {I I^{-1} = A} 

Thus, the nonzero fractional ideals are an abelian group under multiplication.

To see this, note that invertibility is preserved under localization: for a multiplicative set {S}, we have {S^{-1} ( I^{-1} ) = (S^{-1} I)^{-1}}, where the second ideal inverse is with respect to {S^{-1}A}; this follows from the fact that {I} is finitely generated. Note also that invertibility is true for discrete valuation rings.

So for all primes {\mathfrak{p}}, we have {(I I^{-1})_{\mathfrak{p}} = A_{\mathfrak{p}}}, which means the inclusion of {A}-modules {I I^{-1} \rightarrow A} is an isomorphism at each localization. Therefore it is an isomorphism, by general algebra.

The next result says we have unique factorization of ideals:

Theorem 3 (Factorization) Each ideal {I \subset A} can be written uniquely as a product of powers of prime ideals.  

Let’s use the pseudo-inductive argument. Let {I} be the maximal ideal which can’t be written in such a manner, since {A} is Noetherian. Then {I} isn’t prime, so it’s contained in some prime {\mathfrak{p}}. But {I = (I\mathfrak{p}^{-1})\mathfrak{p}}, and {I\mathfrak{p}^{-1} \neq I} can be written as a product of primes, contradiction.

Uniqueness follows by localizing at each prime.

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[...] of the reasons Dedekind domains are so important is Theorem 1 Let be a Dedekind domain with quotient field , a finite separable [...]

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[...] a Dedekind domain with quotient field ; let be a finite separable extension of , and the integral closure of in . [...]

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