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

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If M is a finitely generated module over a commutative ring R generated by elements m1,...,mn with relations

 

then the ith Fitting ideal   of M is generated by the minors (determinants of submatrices) of order   of the matrix  . The Fitting ideals do not depend on the choice of generators and relations of M.

Some authors defined the Fitting ideal   to be the first nonzero Fitting ideal  .

Properties

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The Fitting ideals are increasing

 

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

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If M is free of rank n then the Fitting ideals   are zero for i<n and R for i ≥ n.

If M is a finite abelian group of order   (considered as a module over the integers) then the Fitting ideal   is the ideal  .

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

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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  , the  -module   is coherent, so we may define   as a coherent sheaf of  -ideals; the corresponding closed subscheme of   is called the Fitting image of f.[1][citation needed]

References

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  1. ^ Eisenbud, David; Harris, Joe. The Geometry of Schemes. Springer. p. 219. ISBN 0-387-98637-5.