Fractional ideals

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 25 June 2012

Fractional ideals

Let $A$ be a commutative ring. Let $P$ be a finitely generated projective $A-$module. Let $p\in \mathrm{Spec}\left(A\right).$

• The rank of $P$ at $p$, ${\mathrm{rk}}_p\left(P\right)$ is the rank of the free $A_p-$module $P_p.$
• The rank of $P$ is $n$ if ${\mathrm{rk}}_p\left(P\right)=n$ for all $p\in \mathrm{Spec}\left(A\right).$

The set $$P\left(A\right)=\left\{\genfrac{}{}{0ex}{}{isomorphism\; classes\mathrm{cl}\left(P\right)of}{\genfrac{}{}{0ex}{}{finitely\; generated\; projectiveA-modules}{Pof\; rank\; 1}}\right\}$$ with operation $$\mathrm{cl}\left(M\right)+\mathrm{cl}\left(N\right)=\mathrm{cl}\left(M{\otimes }_{A}N\right)$$ is an abelian group.

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

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 $M\cdot N=A.$

Let $$𝒥=\left\{invertibleA-submodules\; ofB\right\}$$ with product multiplication. Then there is an exact sequence of abelian groups $$\begin{array}{ccccccccccc}\left\{1\right\}& \to & A^{\times }& \to & B^{\times }& \stackrel{\theta }{\to }& 𝒥& \stackrel{\mathrm{cl}}{\to }& P\left(A\right)& \stackrel{\phi }{\to }& P\left(B\right)\\ & & a& \mapsto & a& & & & & & \\ & & & & b& \mapsto & Ab& & & & \\ & & & & & & M& \mapsto & \mathrm{cl}\left(M\right)& \mapsto & \mathrm{cl}\left(B{\otimes }_{A}M\right).\end{array}$$

If $A$ is an integral domain, $S=A-\left\{0\right\}$ and $B$ is the field of fractions of $A$ then $$𝒥\stackrel{\mathrm{cl}}{\to }P\left(A\right)is\; an\; isomorphism.$$

1. An invertible fractional ideal is an element of $𝒥.$
2. A principal fractional ideal is an element of $\mathrm{im}\theta .$

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].

References

References?

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