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 Hans Fitting (1936).
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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(M) of M is generated by the minors (determinants of submatrices) 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(M) to be the first nonzero Fitting ideal Fitti(M).
Properties
The Fitting ideals are increasing
Fitt0(M) ⊆ Fitt1(M) ⊆ Fitt2(M) ...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 Fitti(M) 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 Fitt0(M) 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.