commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...

, the Fitting ideals of a

finitely generated module In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts in ...

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 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'', ). The

Alexander polynomial In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a ve ...

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 definition of scheme-theoretic image of morphisms, which behaves well in families. Given a morphism of schemes

f: X \rightarrow Y

, the Fitting image of ''f'' is defined to be the closed subscheme associated to the sheaf of ideals

Fitt_0(f_* \mathcal_X)

, where

f_* \mathcal_X

is seen as a

\mathcal_Y

-module via the canonical morphism

f^\#

References

* * * *{{Citation , last1=Northcott , first1=D. G. , title=Finite free resolutions , publisher=

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...

, mr=0460383 , year=1976, isbn=978-0-521-60487-1 Commutative algebra