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 .

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 m 1 ,...,m n with relations a_{j1}m_1+\cdots + a_{jn}m_n=0\ (\text{for }j = 1, 2, \dots) then the ith Fitting ideal \operatorname{Fitt}_i(M) 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 \operatorname{Fitt}_i(M) .

Properties

The Fitting ideals are increasing \operatorname{Fitt}_0(M) \subseteq \operatorname{Fitt}_1(M) \subseteq \operatorname{Fitt}_2(M) \subseteq \cdots If M can be generated by n elements then Fitt n (M) = R, and if R is local the converse holds. We have Fitt 0 (M) ⊆ Ann(M) (the annihilator of M), and Ann(M)Fitt i (M) ⊆ Fitt i−1 (M), so in particular if M can be generated by n elements then Ann(M) n  ⊆ Fitt 0 (M).

Examples

If M is free of rank n then the Fitting ideals \operatorname{Fitt}_i(M) are zero for i |M| (considered as a module over the integers) then the Fitting ideal \operatorname{Fitt}_0(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.

References

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