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,...,mn with relations
a j 1 m 1 + ⋯ + a j n m n = 0 ( for j = 1 , 2 , … ) {\displaystyle a_{j1}m_{1}+\cdots +a_{jn}m_{n}=0\ ({\text{for }}j=1,2,\dots )}then the ith Fitting ideal Fitt i ( M ) {\displaystyle \operatorname {Fitt} _{i}(M)} of M is generated by the minors (determinants of submatrices) of order n − i {\displaystyle n-i} of the matrix a j k {\displaystyle 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 ) {\displaystyle I(M)} to be the first nonzero Fitting ideal Fitt i ( M ) {\displaystyle \operatorname {Fitt} _{i}(M)} .
Properties
The Fitting ideals are increasing
Fitt 0 ( M ) ⊆ Fitt 1 ( M ) ⊆ Fitt 2 ( M ) ⊆ ⋯ {\displaystyle \operatorname {Fitt} _{0}(M)\subseteq \operatorname {Fitt} _{1}(M)\subseteq \operatorname {Fitt} _{2}(M)\subseteq \cdots }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 Fitt i ( M ) {\displaystyle \operatorname {Fitt} _{i}(M)} are zero for i<n and R for i ≥ n.
If M is a finite abelian group of order | M | {\displaystyle |M|} (considered as a module over the integers) then the Fitting ideal Fitt 0 ( M ) {\displaystyle \operatorname {Fitt} _{0}(M)} is the ideal ( | M | ) {\displaystyle (|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 → Y {\displaystyle f\colon X\rightarrow Y} , the O Y {\displaystyle {\mathcal {O}}_{Y}} -module f ∗ O X {\displaystyle f_{*}{\mathcal {O}}_{X}} is coherent, so we may define Fitt 0 ( f ∗ O X ) {\displaystyle \operatorname {Fitt} _{0}(f_{*}{\mathcal {O}}_{X})} as a coherent sheaf of O Y {\displaystyle {\mathcal {O}}_{Y}} -ideals; the corresponding closed subscheme of Y {\displaystyle Y} is called the Fitting image of f.1
- Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1, MR 1322960
- Fitting, Hans (1936), "Die Determinantenideale eines Moduls", Jahresbericht der Deutschen Mathematiker-Vereinigung, 46: 195–228, ISSN 0012-0456
- 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, MR 0742853
- Northcott, D. G. (1976), Finite free resolutions, Cambridge University Press, ISBN 978-0-521-60487-1, MR 0460383
References
Eisenbud, David; Harris, Joe. The Geometry of Schemes. Springer. p. 219. ISBN 0-387-98637-5. 0-387-98637-5 ↩