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

${\displaystyle a_{j1}m_{1}+\cdots +a_{jn}m_{n}=0\ ({\text{for }}j=1,2,\dots )\,}$

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.

Fitting image

The zeroth Fitting ideal can be used also to give a definition of scheme-theoretic image of morphisms, which behaves well in families. Given a morphism of schemes ${\displaystyle f:X\rightarrow Y}$, the Fitting image of f is defined to be the closed subscheme associated to the sheaf of ideals ${\displaystyle Fitt_{0}(f_{*}{\mathcal {O}}_{X})}$, where ${\displaystyle f_{*}{\mathcal {O}}_{X}}$ is seen as a ${\displaystyle {\mathcal {O}}_{Y}}$-module via the canonical morphism ${\displaystyle f^{\#}}$.

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 0742853
• Northcott, D. G. (1976), Finite free resolutions, Cambridge University Press, ISBN 978-0-521-60487-1, MR 0460383