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.

Definition
If M is a finitely generated module over a commutative ring R generated by elements m1,...,mn 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 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&minus;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.