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 .
If M is a finitely generated module over a commutative ring R generated by elements m1,...,mnwith relations
aj1m1+ … +ajnmn=0 (forj=1,2,...)
then the ith Fitting ideal
\operatorname{Fitt}i(M)
n-i
ajk
Some authors defined the Fitting ideal
I(M)
\operatorname{Fitt}i(M)
The Fitting ideals are increasing
\operatorname{Fitt}0(M)\subseteq\operatorname{Fitt}1(M)\subseteq\operatorname{Fitt}2(M)\subseteq …
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-1(M), so in particular if M can be generated by n elements then Ann(M)n ⊆ Fitt0(M).
If M is free of rank n then the Fitting ideals
\operatorname{Fitt}i(M)
If M is a finite abelian group of order
|M|
\operatorname{Fitt}0(M)
(|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.
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\colonX → Y
l{O}Y
f*l{O}X
\operatorname{Fitt}0(f*l{O}X)
l{O}Y
Y