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 ${\displaystyle \operatorname {Fitt} _{i}(M)}$ of M is generated by the minors (determinants of submatrices) of order ${\displaystyle n-i}$ of the matrix ${\displaystyle a_{jk}}$. The Fitting ideals do not depend on the choice of generators and relations of M.

Some authors defined the Fitting ideal ${\displaystyle I(M)}$ to be the first nonzero Fitting ideal ${\displaystyle \operatorname {Fitt} _{i}(M)}$.

Properties

The Fitting ideals are increasing

${\displaystyle \operatorname {Fitt} _{0}(M)\subseteq \operatorname {Fitt} _{1}(M)\subseteq \operatorname {Fitt} _{2}(M)\subseteq \cdots }$

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 ${\displaystyle \operatorname {Fitt} _{i}(M)}$ are zero for i<n and R for i ≥ n.

If M is a finite abelian group of order ${\displaystyle |M|}$ (considered as a module over the integers) then the Fitting ideal ${\displaystyle \operatorname {Fitt} _{0}(M)}$ is the ideal ${\displaystyle (|M|)}$.

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 ${\displaystyle f\colon X\rightarrow Y}$, the ${\displaystyle {\mathcal {O}}_{Y}}$-module ${\displaystyle f_{*}{\mathcal {O}}_{X}}$ is coherent, so we may define ${\displaystyle \operatorname {Fitt} _{0}(f_{*}{\mathcal {O}}_{X})}$ as a coherent sheaf of ${\displaystyle {\mathcal {O}}_{Y}}$-ideals; the corresponding closed subscheme of ${\displaystyle Y}$ is called the Fitting image of f.^{[1][citation needed]}