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 of M is generated by the minors (determinants of submatrices) of order n-i of the matrix a_{jk}. The Fitting ideals do not depend on the choice of generators and relations of M. Some authors defined the Fitting ideal I(M) to be the first nonzero Fitting ideal.

Properties

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

Examples

If M is free of rank n then the Fitting ideals are zero for i<n and R for i ≥ n. If M is a finite abelian group of order |M| (considered as a module over the integers) then the Fitting ideal is the ideal (|M|). 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 variant of the notion of scheme-theoretic image of a morphism, a variant that behaves well in families. Specifically, given a finite morphism of noetherian schemes, the -module is coherent, so we may define as a coherent sheaf of -ideals; the corresponding closed subscheme of Y is called the Fitting image of f.