Fitting ideal


In commutative algebra, the Fitting ideals of a finitely generated module over a commutative ring describe the obstructions to generating the module by a given number of elements. They were introduced by.

Definition

If M is a finitely generated module over a commutative ring R generated by elements m1,...,mn
with relations
then the ith Fitting ideal Fitti of M is generated by the minors of order n − i of the matrix ajk.
The Fitting ideals do not depend on the choice of generators and relations of M.
Some authors defined the Fitting ideal I to be the first nonzero Fitting ideal Fitti.

Properties

The Fitting ideals are increasing
If M can be generated by n elements then Fittn = R, and if R is local the converse holds. We have Fitt0 ⊆ Ann, and AnnFitti ⊆ Fitti−1, so in particular if M can be generated by n elements then Annn ⊆ Fitt0.

Examples

If M is free of rank n then the Fitting ideals Fitti are zero for i<n and R for i ≥ n.
If M is a finite abelian group of order |M| then the Fitting ideal Fitt0 is the ideal .
The Alexander polynomial of a knot is a generator of the Fitting ideal of the first homology of the infinite abelian cover of the knot complement.

Fitting image

The zeroth Fitting ideal can be used also to give a definition of scheme-theoretic image of morphisms, which behaves well in families. Given a morphism of schemes, the Fitting image of f is defined to be the closed subscheme associated to the sheaf of ideals, where is seen as a -module via the canonical morphism.