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,...,mn with relations
- [math]\displaystyle{ a_{j1}m_1+\cdots + a_{jn}m_n=0\ (\text{for }j = 1, 2, \dots) }[/math]
then the ith Fitting ideal [math]\displaystyle{ \operatorname{Fitt}_i(M) }[/math] of M is generated by the minors (determinants of submatrices) of order [math]\displaystyle{ n-i }[/math] of the matrix [math]\displaystyle{ a_{jk} }[/math]. The Fitting ideals do not depend on the choice of generators and relations of M.
Some authors defined the Fitting ideal [math]\displaystyle{ I(M) }[/math] to be the first nonzero Fitting ideal [math]\displaystyle{ \operatorname{Fitt}_i(M) }[/math].
Properties
The Fitting ideals are increasing
- [math]\displaystyle{ \operatorname{Fitt}_0(M) \subseteq \operatorname{Fitt}_1(M) \subseteq \operatorname{Fitt}_2(M) \subseteq \cdots }[/math]
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 [math]\displaystyle{ \operatorname{Fitt}_i(M) }[/math] are zero for i<n and R for i ≥ n.
If M is a finite abelian group of order [math]\displaystyle{ |M| }[/math] (considered as a module over the integers) then the Fitting ideal [math]\displaystyle{ \operatorname{Fitt}_0(M) }[/math] is the ideal [math]\displaystyle{ (|M|) }[/math].
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 [math]\displaystyle{ f \colon X \rightarrow Y }[/math], the [math]\displaystyle{ \mathcal{O}_Y }[/math]-module [math]\displaystyle{ f_* \mathcal{O}_X }[/math] is coherent, so we may define [math]\displaystyle{ \operatorname{Fitt}_0(f_* \mathcal{O}_X) }[/math] as a coherent sheaf of [math]\displaystyle{ \mathcal{O}_Y }[/math]-ideals; the corresponding closed subscheme of [math]\displaystyle{ Y }[/math] is called the Fitting image of f.[1][citation needed]
References
- ↑ Eisenbud, David; Harris, Joe. The Geometry of Schemes. Springer. pp. 219. ISBN 0-387-98637-5. https://link.springer.com/book/10.1007/b97680.
- 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
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