Fitting ideal explained

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,...,mnwith relations

aj1m1+ … +ajnmn=0 (forj=1,2,...)

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

ajk

. 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

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).

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\colonXY

, the

l{O}Y

-module

f*l{O}X

is coherent, so we may define

\operatorname{Fitt}0(f*l{O}X)

as a coherent sheaf of

l{O}Y

-ideals; the corresponding closed subscheme of

Y

is called the Fitting image of f.[1]

Notes and References

  1. Book: Eisenbud. David. David Eisenbud. Harris. Joe . Joe Harris (mathematician). The Geometry of Schemes . . 0-387-98637-5 . 219 .