Fitting ideals control fibre generator loci

The fibre of a presented module: let $U=\mathrm{Spec}A$ be an affine open with $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$ for a finitely generated $A$-module $M$, and fix a presentation $A^{(J)}\stackrel{\phi }{\to }{A}^n\to M\to 0$ with $n$ finite; for $\mathfrak{p}\in U$ the localisation and the tensor product with $\kappa (\mathfrak{p})=A_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{A}_{\mathfrak{p}}$ give the exact sequence $\kappa (\mathfrak{p}{)}^{(J)}\stackrel{\phi \otimes \kappa (\mathfrak{p})}{\to }\kappa (\mathfrak{p}{)}^n\to M{\otimes }_A\kappa (\mathfrak{p})\to 0$, and $\mathcal{F}(\mathfrak{p})=M_{\mathfrak{p}}{\otimes }_{A_{\mathfrak{p}}}\kappa (\mathfrak{p})=M{\otimes }_A\kappa (\mathfrak{p})$; hence, with $\phi$ also denoting its reduced matrix, ${\dim }_{\kappa (\mathfrak{p})}\mathcal{F}(\mathfrak{p})=n-\mathrm{rank}(\phi \otimes \kappa (\mathfrak{p}))$.

Minors detect the rank drop: for a given $r\ge 0$ and $s=n-r$, the inequality $\mathrm{rank}(\phi \otimes \kappa (\mathfrak{p}))<s$ holds if and only if all $s\times s$ minors of the reduced matrix vanish, which happens if and only if every $s\times s$ minor of the matrix of $\phi$ lies in $\mathfrak{p}$, that is, $I_{n-r}(\phi )\subseteq \mathfrak{p}$; the conventions of [F1] give $I_{n-r}(\phi )=A$ when $s\le 0$, and $\mathrm{rank}<s$ is then impossible, while for $s>n$ one has $I_{n-r}(\phi )=0$ and $\mathrm{rank}<s$ holds always.

The rank locus on a chart: combining steps 1.1 and 2.1, for $\mathfrak{p}\in U$ one has ${\dim }_{\kappa (\mathfrak{p})}\mathcal{F}(\mathfrak{p})>r$ if and only if $I_{n-r}(\phi )\subseteq \mathfrak{p}$; by [F1] the stalk of ${\mathrm{Fitt}}_r(\mathcal{F})$ at $\mathfrak{p}$ is $I_{n-r}(\phi )\cdot A_{\mathfrak{p}}$, and $I\cdot A_{\mathfrak{p}}\subseteq \mathfrak{p}{A}_{\mathfrak{p}}$ if and only if $I\subseteq \mathfrak{p}$; hence $\mathfrak{p}\in V({\mathrm{Fitt}}_r(\mathcal{F}))$ if and only if ${\dim }_{\kappa (\mathfrak{p})}\mathcal{F}(\mathfrak{p})>r$.

Converse, the case $n=r$: let $U=\mathrm{Spec}A$ be affine with $\mathcal{F}{\mathrm{\mid }}_U\cong \tilde{M}$, presentation $A^{(J)}\stackrel{\phi }{\to }{A}^n\to M\to 0$, and assume $I_{n-r}(\phi )=A$, $I_{n-r+1}(\phi )=0$ for some $r\ge 1$. If $A=0$ the conclusion is vacuous, so assume $A\mathrm{\ne }0$; then $n\ge r$, since $n-r+1\le 0$ would force $A=I_{n-r+1}(\phi )=0$. If $n=r$, then $I_1(\phi )=I_{n-r+1}(\phi )=0$ says that every entry of the matrix of $\phi$ is zero, so $\phi =0$ and $M=\mathrm{coker}\phi \cong A^n=A^r$ is free of rank $r$.

Claim 1 globally: the affine charts of step 3.1 cover $X$ by [F2] and the assertion is pointwise in each chart, so $V({\mathrm{Fitt}}_r(\mathcal{F}))=\{x:{\dim }_{\kappa (x)}\mathcal{F}(x)>r\}$ and $D({\mathrm{Fitt}}_r(\mathcal{F}))$ is its complement $\{x:{\dim }_{\kappa (x)}\mathcal{F}(x)\le r\}$; this proves claim 1.

Converse, the case $n>r$: keep the hypotheses of step 3.2 with $s=n-r\ge 1$. The $s\times s$ minors of $\phi$ generate $I_s(\phi )=A$, so the distinguished opens $D(\delta )\subseteq \mathrm{Spec}A$, as $\delta$ runs over the $s\times s$ minors, cover $\mathrm{Spec}A$; fix such a minor $\delta$ and invert it. Over $A_{\delta }$ the selected minor is a unit, so after permuting the $s$ chosen rows and columns into the top-left corner, row operations by invertible $n\times n$ matrices and column operations by an automorphism of $A_{\delta }^{(J)}$ put the matrix of $\phi \otimes A_{\delta }$ into block form $\left(\begin{array}{cc}{I}_s& 0\\ C& \psi \end{array}\right)$ with $\psi$ of size $r$ rows; the $(s+1)\times (s+1)$ minors formed from $s$ rows and $s$ columns of the identity block together with one further row and column are, up to sign, exactly the entries of $\psi$, and they vanish because $I_{s+1}(\phi )\otimes A_{\delta }=I_{n-r+1}(\phi )\otimes A_{\delta }=0$; hence $\psi =0$ and $M_{\delta }\cong \mathrm{coker}\left(\begin{array}{cc}{I}_s& 0\\ C& 0\end{array}\right)\cong A_{\delta }^r$, because the image is the graph $\{(x,Cx)\}$, a direct summand of $A_{\delta }^n$ of rank $s$. Thus $\mathcal{F}$ is free of rank $r$ on each member $D(\delta )$ of a cover of $U$, so $\mathcal{F}{\mathrm{\mid }}_U$ is locally free of rank $r$.

Claim 2, the support: taking $r=0$ in claim 1 gives $V({\mathrm{Fitt}}_0(\mathcal{F}))=\{x:{\dim }_{\kappa (x)}\mathcal{F}(x)>0\}=\{x:\mathcal{F}(x)\mathrm{\ne }0\}$. For finite type $\mathcal{F}$ one has $\mathcal{F}(x)=0$ only if $\mathcal{F}$ vanishes on a neighbourhood of $x$, hence only if ${\mathcal{F}}_x=0$, and conversely ${\mathcal{F}}_x=0$ gives $\mathcal{F}(x)=0$; therefore $\{x:\mathcal{F}(x)\mathrm{\ne }0\}=\{x:{\mathcal{F}}_x\mathrm{\ne }0\}=\mathrm{Supp}(\mathcal{F})$, and this agrees with the closedness of the support in [F5].

Claim 3, the locally free case: suppose $\mathcal{F}$ is finite locally free of rank $r$, so that by [F2] $X$ is covered by affine opens $U=\mathrm{Spec}A$ with $\mathcal{F}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r=\widetilde{A^r}$; for the presentation $A^r=A^r$ with $n=r$ generators and no relations the matrix has $r$ rows and $0$ columns, so by the conventions of [F1] $I_{r-k}(\phi )=A$ for $k\ge r$ and $I_{r-k}(\phi )=0$ for $0\le k<r$, whence ${\mathrm{Fitt}}_k(\mathcal{F}){\mathrm{\mid }}_U={\mathcal{O}}_U$ for $k\ge r$ and $=0$ for $k<r$; in particular ${\mathrm{Fitt}}_{r-1}(\mathcal{F}){\mathrm{\mid }}_U=0$ and ${\mathrm{Fitt}}_r(\mathcal{F}){\mathrm{\mid }}_U={\mathcal{O}}_U$ for every member of a cover of $X$, and since membership in an ideal subsheaf and the equality with ${\mathcal{O}}_X$ are checked on a cover, ${\mathrm{Fitt}}_{r-1}(\mathcal{F})=0$ and ${\mathrm{Fitt}}_r(\mathcal{F})={\mathcal{O}}_X$.

Claim 4 globally: let $U\subseteq X$ be open with ${\mathrm{Fitt}}_{r-1}(\mathcal{F}){\mathrm{\mid }}_U=0$ and ${\mathrm{Fitt}}_r(\mathcal{F}){\mathrm{\mid }}_U={\mathcal{O}}_U$. If $r=0$, claim 1 gives ${\dim }_{\kappa (x)}\mathcal{F}(x)\le 0$, hence $\mathcal{F}(x)=0$, for every $x\in U$, and [F4] then shows that $\mathcal{F}$ vanishes on a neighbourhood of every point of $U$, so that $\mathcal{F}{\mathrm{\mid }}_U=0$ is locally free of rank $0$. If $r\ge 1$, cover $U$ by affine charts $V=\mathrm{Spec}A$ with $\mathcal{F}{\mathrm{\mid }}_V\cong \tilde{M}$; on each chart the restriction of the hypotheses reads $I_{n-r}(\phi )=A$ and $I_{n-r+1}(\phi )=0$ by [F1], so steps 3.2 and 4.2 show that $\mathcal{F}{\mathrm{\mid }}_V$ is free of rank $r$ on the members of a cover of $V$; these covers for all charts combine to a cover of $U$ by opens on which $\mathcal{F}$ is free of rank $r$. This proves claim 4.

Claim 5, sharpness: let $k$ be a field, $A=k[t]\mathrm{/}(t^2)$ with class $\epsilon$ of $t$, so that ${\epsilon }^2=0$ and $\epsilon \mathrm{\ne }0$, and let $M=A\mathrm{/}(\epsilon )=k$; then $M$ is generated by one element with the single relation $\epsilon \cdot 1=0$, giving a presentation $A\stackrel{\epsilon }{\to }A\to M\to 0$, and $\mathcal{F}=\tilde{M}$ is a finite type quasi-coherent module on $X=\mathrm{Spec}A$, whose underlying space is the single point $\mathfrak{p}=(\epsilon )$. The fibre there is $\mathcal{F}(\mathfrak{p})=M{\otimes }_A\kappa (\mathfrak{p})=M\mathrm{/}\epsilon M=k$, of dimension $1$ over $\kappa (\mathfrak{p})=k$; the Fitting ideals of this presentation are ${\mathrm{Fitt}}_0(M)=I_1(\phi )=(\epsilon )\mathrm{\ne }0$ and ${\mathrm{Fitt}}_1(M)=I_0(\phi )=A$. By the contrapositive of claim 3 proved in step 5.2, a locally free module of rank $1$ would have ${\mathrm{Fitt}}_0=0$, so $\mathcal{F}$ is not locally free of rank $1$; directly, $\mathcal{F}\cong {\mathcal{O}}_X$ would give $M\cong \mathrm{\Gamma }(X,{\mathcal{O}}_X)=A$, impossible because ${\dim }_{k}M=1$ while ${\dim }_{k}A=2$. This proves claim 5 and shows that the two Fitting equalities, not merely the pointwise fibre dimension, are needed for local freeness.

Conclusion and choice accounting: claims 1 and 2 are steps 4.1 and 5.1, claim 3 is step 5.2, claim 4 is step 5.3 and claim 5 is step 6.1. All constructions are local and finite: a presentation with finitely many generators is used on one affine chart at a time, the minors of a matrix are indexed by finite subsets of that presentation, and the cover $\{D(\delta )\}$ in step 4.2 is the set of all distinguished opens defined by the minors, so no selection of points, presentations or minors is made; the Axiom of Choice is inherited from the associated-sheaf existence theorem used by the Fitting definition [F1], and no Noetherian hypothesis is used.