Fractional ideals

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 25 June 2012

Fractional ideals

Let A be a commutative ring. Let P be a finitely generated projective A-module. Let pSpec(A).

The set P(A) = { isomorphism classes cl(P) of finitely generated projective A-modules P of rank 1 } with operation cl(M)+cl(N) = cl( MAN ) is an abelian group.

Let A be a commutative ring, S a multiplicatively closed subset such that if sS then s is not a zero divisor in A, B=S-1A.

An invertible sub A-module is a sub A-module M of B such that there exists a sub A-module N of B such that MN=A.

Let 𝒥 = { invertible A-submodules of B } with product multiplication. Then there is an exact sequence of abelian groups {1} A× B× θ 𝒥 cl P(A) φ P(B) a a b Ab M cl(M) cl(BAM).

If A is an integral domain, S=A-{0} and B is the field of fractions of A then 𝒥clP(A) is an isomorphism.

  1. An invertible fractional ideal is an element of 𝒥.
  2. A principal fractional ideal is an element of imθ.

Notes and References

The Theorem is given in [Bou, Comm. Ch.II, §5 no.4, Proposition 7] and the definitions and proofs of well-definedness of rank of P are given in [Bou, Comm. Ch.II, §5 no.3].



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