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 Hans Fitting (1936).

Definition

If M is a finitely generated module over a commutative ring R generated by elements m₁,...,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 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 Fitt_i(M).

Properties

The Fitting ideals are increasing

Fitt₀(M) ⊆ Fitt₁(M) ⊆ Fitt₂(M) ...

If M can be generated by n elements then Fitt_n(M) = R, and if R is local the converse holds. We have Fitt₀(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 ⊆ Fitt₀(M).

Examples

If M is free of rank n then the Fitting ideals Fitt_i(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.

References

Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1, MR 1322960
Fitting, Hans (1936), "Die Determinantenideale eines Moduls", Jahresbericht der Deutschen Mathematiker-Vereinigung, 46: 195–228, ISSN 0012-0456
Mazur, Barry; Wiles, Andrew (1984), "Class fields of abelian extensions of Q", Inventiones Mathematicae, 76 (2): 179–330, ISSN 0020-9910, MR 742853, doi:10.1007/BF01388599
Northcott, D. G. (1976), Finite free resolutions, Cambridge University Press, ISBN 978-0-521-60487-1, MR 0460383

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