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 ''m''₁,...,''m''_''n''
with relations

:$a_m_1+\cdots + a_m_n=0\ (\textj = 1, 2, \dots)$

then the ''i''th Fitting ideal $\operatorname_i(M)$ of ''M'' is generated by the minors (determinants of submatrices) of order $n-i$ of the matrix $a_$.
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 $\operatorname_i(M)$.

Properties

The Fitting ideals are increasing

: $\operatorname_0(M) \subseteq \operatorname_1(M) \subseteq \operatorname_2(M) \subseteq \cdots$

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 $\operatorname_i(M)$ 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 $\operatorname_0(M)$ 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 $\mathcal_Y$-module $f_* \mathcal_X$ is coherent, so we may define $\operatorname_0(f_* \mathcal_X)$ as a coherent sheaf of $\mathcal_Y$-ideals; the corresponding closed subscheme of $Y$ is called the Fitting image of ''f''.

References

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Commutative algebra

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