Fitting ideal

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

with relations

:$a\_\{j1\}m\_1+\backslash cdots\; +\; a\_\{jn\}m\_n=0\backslash \; (\backslash text\{for\; \}j\; =\; 1,\; 2,\; \backslash dots)$

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

The Fitting ideals are increasing

: $\backslash operatorname\{Fitt\}\_0(M)\; \backslash subseteq\; \backslash operatorname\{Fitt\}\_1(M)\; \backslash subseteq\; \backslash operatorname\{Fitt\}\_2(M)\; \backslash subseteq\; \backslash cdots$

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).

If M is free of rank n then the Fitting ideals $\backslash operatorname\{Fitt\}\_i(M)$ 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 $\backslash operatorname\{Fitt\}\_0(M)$ 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.

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 $f\; \backslash colon\; X\; \backslash rightarrow\; Y$, the $\backslash mathcal\{O\}\_Y$-module $f\_*\; \backslash mathcal\{O\}\_X$ is coherent, so we may define $\backslash operatorname\{Fitt\}\_0(f\_*\; \backslash mathcal\{O\}\_X)$ as a coherent sheaf of $\backslash mathcal\{O\}\_Y$-ideals; the corresponding closed subscheme of $Y$ is called the Fitting image of f.{{cite book |last1=Eisenbud|first1=David|author1-link=David Eisenbud|last2=Harris| first2=Joe |author2-link=Joe Harris (mathematician)|title=The Geometry of Schemes |publisher=Springer |isbn=0-387-98637-5 |pages=219 |url=https://link.springer.com/book/10.1007/b97680}}{{fact|date=August 2023}}

{{Reflist}}

• {{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1 | mr=1322960 | year=1995 | volume=150}}
• {{Citation | last=Fitting | first=Hans | authorlink=Hans Fitting| title=Die Determinantenideale eines Moduls | url=http://resolver.sub.uni-goettingen.de/purl?PPN37721857X | year=1936 | journal=Jahresbericht der Deutschen Mathematiker-Vereinigung | issn=0012-0456 | volume=46 | pages=195–228}}
• {{Citation | last1=Mazur | first1=Barry | author1-link=Barry Mazur | last2=Wiles | first2=Andrew | author2-link=Andrew Wiles | title=Class fields of abelian extensions of Q | doi=10.1007/BF01388599 | mr=742853 | year=1984 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=76 | issue=2 | pages=179–330}}
• {{Citation | last1=Northcott | first1=D. G. | title=Finite free resolutions | publisher=Cambridge University Press | mr=0460383 | year=1976|isbn=978-0-521-60487-1}}

Category:Commutative algebra

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