Fitting ideal

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

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

$a_{j1}{m}_1+\cdots +a_{jn}{m}_n=0(\text{for }j=1,2,\dots )$

then the ith Fitting ideal ${\mathrm{Fitt}}_i(M)$ 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 ${\mathrm{Fitt}}_i(M)$.

Properties

The Fitting ideals are increasing

${\mathrm{Fitt}}_0(M)\subseteq {\mathrm{Fitt}}_1(M)\subseteq {\mathrm{Fitt}}_2(M)\subseteq \cdots$

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 ${\mathrm{Fitt}}_i(M)$ 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 ${\mathrm{Fitt}}_0(M)$ 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 $f:X\to Y$, the ${{𝒪}}_Y$-module $f_{*}{{𝒪}}_X$ is coherent, so we may define ${\mathrm{Fitt}}_0(f_{*}{{𝒪}}_X)$ as a coherent sheaf of ${{𝒪}}_Y$-ideals; the corresponding closed subscheme of $Y$ is called the Fitting image of f.

References

• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1 
• Fitting, Hans (1936), "Die Determinantenideale eines Moduls", Jahresbericht der Deutschen Mathematiker-Vereinigung 46: 195–228, ISSN 0012-0456, http://resolver.sub.uni-goettingen.de/purl?PPN37721857X 
• Mazur, Barry; Wiles, Andrew (1984), "Class fields of abelian extensions of Q", Inventiones Mathematicae 76 (2): 179–330, doi:10.1007/BF01388599, ISSN 0020-9910 
• Northcott, D. G. (1976), Finite free resolutions, Cambridge University Press, ISBN 978-0-521-60487-1