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,...,mn with relations

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

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.

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