Fitting ideal sheaves

Assume the Axiom of Choice (The Axiom of Choice), as used by the associated-module-sheaf existence theorem below. Let $X$ be a scheme and let $\mathcal{F}$ be a quasi-coherent ${\mathcal{O}}_X$-module of finite type (Finite type and finitely presented module sheaves). Let $k\ge 0$ be an integer. The $k$-th Fitting ideal sheaf ${\mathrm{Fitt}}_k(\mathcal{F})$ is defined affine locally as follows.

Let $U=\mathrm{Spec}A$ be affine and let $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$ for a finitely generated $A$-module $M$; choose a finite generating set and the resulting presentation $A^{(J)}\stackrel{\phi }{\to }{A}^n\stackrel{}{\to }M⟶0$ with $n$ finite and $J$ an arbitrary index set (Finitely presented modules and finitely presented algebras). Let $I_{n-k}(\phi )$ be the ideal of $A$ generated by the $(n-k)\times (n-k)$ minors of a matrix of $\phi$, with the conventions $I_r(\phi )=A$ for $r\le 0$ and $I_r(\phi )=0$ once $r$ exceeds the number of rows or the number of columns (Fitting ideals do not depend on a presentation). Then ${\mathrm{Fitt}}_k(\mathcal{F}){\mathrm{\mid }}_U:=\widetilde{I_{n-k}(\phi )},$ the associated sheaf of that ideal, viewed as a subsheaf of ${\mathcal{O}}_U=\tilde{A}$ (Module sheaf on an affine scheme). By Fitting ideals do not depend on a presentation the ideal $I_{n-k}(\phi )$ depends only on $M$ and $k$, not on the chosen generating set, presentation or integer $n$, and it commutes with localisation; hence the local data are canonical and compatible on overlaps $D(f)$, where they are given by $I_{n-k}(\phi )\cdot A_f$ (Localisation of modules is exact, The associated module sheaf exists). Gluing over an affine cover of $X$ produces an ideal sheaf ${\mathrm{Fitt}}_k(\mathcal{F})\subseteq {\mathcal{O}}_X\mathrm{.}$ It is quasi-coherent as an ideal sheaf (Quasi-coherent ideal sheaves), since on each affine chart it is the ideal sheaf associated to the $A$-module $I_{n-k}(\phi )$. The associated-sheaf existence and identification on distinguished opens use the stated Axiom of Choice through The associated module sheaf exists; presentation independence and localisation of minors use only the cited algebraic supplier.

Boundary values. The convention $I_r=A$ for $r\le 0$ records the empty product, a $0\times 0$ minor being $1$. For $k=-1$ the convention $I_r=0$ for $r$ larger than the number of rows gives ${\mathrm{Fitt}}_{-1}(\mathcal{F})=0$. For $\mathcal{F}=0$ the presentation with $n=0$ gives ${\mathrm{Fitt}}_k(0)={\mathcal{O}}_X$ for every $k\ge 0$. In general ${\mathrm{Fitt}}_0$ is generated by the $n\times n$ minors of a relation matrix, so it need not be ${\mathcal{O}}_X$: if $\mathcal{F}$ is free of rank $r$ on $U$, presented there with $n=r$ generators and no relations, then ${\mathrm{Fitt}}_k(\mathcal{F}){\mathrm{\mid }}_U=0$ for $k<r$ and ${\mathrm{Fitt}}_k(\mathcal{F}){\mathrm{\mid }}_U={\mathcal{O}}_U$ for $k\ge r$; in particular ${\mathrm{Fitt}}_0({\mathcal{O}}_U)=0$. For $0\le k<n$, the ideal ${\mathrm{Fitt}}_k(\mathcal{F}){\mathrm{\mid }}_U$ is zero precisely when all $(n-k)\times (n-k)$ minors of the chosen relation matrix vanish. For $k\ge n$ it is ${\mathcal{O}}_U$ by convention, so it is zero only when $A$ is the zero ring.

Basic properties. ${\mathrm{Fitt}}_k(\mathcal{F})$ is defined for every $k\ge 0$ and depends only on $\mathcal{F}$; it is compatible with base change and with restriction to open subschemes, and for $k=0$ it cuts out the support in the finite type case (the rank-locus theorem on this page). The ideals are nested in the sense $I_r\subseteq I_s$ for $s\le r$, since a larger minor is an $A$-linear combination of smaller ones by Laplace expansion; hence ${\mathrm{Fitt}}_{k-1}\subseteq {\mathrm{Fitt}}_k$ for all $k\ge 1$, so the Fitting ideals increase with $k$ and reach ${\mathrm{Fitt}}_n(\mathcal{F})={\mathcal{O}}_X$ on a chart with $n$ generators.