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 $\operatorname {Fitt} _{i}(M)$ 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 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 $\operatorname {Fitt} _{i}(M)$ 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 $\operatorname {Fitt} _{0}(M)$ 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.

Fitting image

The zeroth Fitting ideal can be used also to give a variant of the notion of scheme-theoretic image of a morphism, a variant that behaves well in families. Specifically, given a finite morphism of noetherian schemes $f\colon X\rightarrow Y$, the ${\mathcal {O}}_{Y}$-module $f_{*}{\mathcal {O}}_{X}$ is coherent, so we may define $\operatorname {Fitt} _{0}(f_{*}{\mathcal {O}}_{X})$ as a coherent sheaf of ${\mathcal {O}}_{Y}$-ideals; the corresponding closed subscheme of $Y$ is called the Fitting image of f.

References

• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, vol. 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