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, doi:10.1007/BF01388599, ISSN 0020-9910, MR 742853
• Northcott, D. G. (1976), Finite free resolutions, Cambridge University Press, ISBN 978-0-521-60487-1, MR 0460383