fractional ideal

Fractional Ideal

Idea: get methods that work for primes \(p\in {\mathbf{Z}}\) to work for ideals \(\left\langle{p}\right\rangle \in \operatorname{Id}({\mathbf{Z}})\), and then generalize to other rings e.g. to get unique factorization. attachments/Pasted%20image%2020220123202618.png Definitions:

Colon Ideal

attachments/Pasted%20image%2020220123195354.png

Invertibility

Invertibility of fractional ideals: attachments/Pasted%20image%2020220123195729.png

Fact: in a Dedekind domain every nonzero fractional ideal is invertible.

Example of noninvertible ideals: attachments/Pasted%20image%2020220123195920.png

Primality

attachments/Pasted%20image%2020220124114019.png # Exercises

Let \(A\) be an integral domain with fraction field \(K\) and let \(M\) be a nonzero A-submodule of \(K\). Then \(M^{\vee} \simeq(A: M):=\{x \in K: x M \subseteq A\}\); in particular, if \(M\) is an invertible fractional ideal then \(M^{\vee} \simeq M^{-1}\) and \(M^{\vee \vee} \simeq M\).

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class group

For a Noetherian domain \(R\): the ideal group \({ \ddot{\operatorname{Id}} }^{\times}(R)\) is the abelian group of invertible fractional ideals. The elements are not necessarily ideals. - Uses that \({ \ddot{\operatorname{Id}} }(R)\) is commutative and associative under multiplication with unit \(R = \left\langle{1}\right\rangle\), so the subset of invertible elements forms a commutative group.
class field theory

See class group, fractional ideal, Galois module, totally unramified extension, Artin map.
Weil divisor

Equivalently, \(\mathop{\mathrm{Cart}}\operatorname{Div}(X)\) is the group of invertible fractional ideals.
Commutative Algebra: Definitions

What is a colon ideal?

What is a principal fractional ideal?

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