Definition
If ''M'' is a finitely generated module over a commutative ring ''R'' generated by elements ''m''1,...,''m''''n'' with relations : then the ''i''th Fitting ideal 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 Fitt''i''(''M'').Properties
The Fitting ideals are increasing : Fitt0(''M'') ⊆ Fitt1(''M'') ⊆ Fitt2(''M'') ... 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 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 Fitt0(''M'') is the ideal (, ''M'', ). TheFitting 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 , the Fitting image of ''f'' is defined to be the closed subscheme associated to the sheaf of ideals , where is seen as a -module via the canonical morphism .References
* * * *{{Citation , last1=Northcott , first1=D. G. , title=Finite free resolutions , publisher=