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Let $(X,\mathcal{O}_X)$ be a quasi-compact ringed space which is locally isomorphic to an affine variety. Let $x\in X$, $\mathcal{F}$ be a coherent sheaf on $X$, and $n$ be the minimum number of generators of $\mathcal{F}_x$ as an $\mathcal{O}_{X,x}$-module.

For any open affine $U\subseteq X$ we have $\mathcal{O}_{X,x}\cong A_{\mathfrak{m}_x}$ as rings, and $\mathcal{G}_x\cong M_{\mathfrak{m}_x}$ as $A_{\mathfrak{m}_x}$-modules, where $M=\mathcal{F}(U)$, $A=\Gamma(U,\mathcal{O}_X)$, $\mathfrak{m}_x=\{f\in A\mid f(x)=0\}$, and $\mathcal{G}=\mathcal{F}\mid_U$. Let $m$ be the minimum number of generators of $M_{\mathfrak{m}_x}$ as an $A_{\mathfrak{m}_x}$-module.

Then is $m=n$?

My confusion is in relating $\mathcal{F}_x$ and $\mathcal{G}_x$. Would it follow that $\mathcal{F}_x\cong\mathcal{G}_x$ as abelian groups because for any sheaf $\mathcal{H}$ we have $$\mathcal{H}_x=\mathop{\lim_{\longrightarrow}}_{V\ni x}\mathcal{H}(V)$$ and since $\mathcal{G}=\mathcal{F}\mid_U$ it doesn’t matter where we “start” because we are passing to smaller and smaller sets anyway? If so, how could I make this argument formally?

I’m still not sure we can then conclude that $m=n$, since the actions might not be compatible with the isomorphism of abelian groups.

Any help would be much appreciated.

Update: Many thanks for the advice in the comments, I have an affirmative answer to the following statement in this question here:

Suppose we have a directed set $\langle I,\leq\rangle$, with a direct system $\langle A_i,f_{ij}\rangle$ of rings and a direct system $\langle M_i,g_{ij}\rangle$ of abelian groups, such that each $M_i$ is an $A_i$ module via $h_i:A_i\times M_i\to M_i$. Then suppose that these actions are compatible with the direct system, so $$g_{ij}(h_i(a,m))=h_j(f_{ij}(a),g_{ij}(m))$$ Then we have an action $$\mathop{\lim_{\longrightarrow}}A_i\times\mathop{\lim_{\longrightarrow}}M_i\to\mathop{\lim_{\longrightarrow}}M_i$$ which is determined by this system.

Then I think this proves the result. Since our sets are cofinal, we'll get isomorphic rings and abelian groups with the same action inherited from the direct systems, and so they should have the same dimension as modules?

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  • Do you mean $\mathcal{O}{X,x}$? In this case you should try to prove that $A{\mathfrak{m}x}$ be isomorphic to $\mathcal{O}{X,x}$. – Pol van Hoften Jul 02 '19 at 12:49
  • @user45878 Yes, by $\mathcal{O}x$ I mean $\mathcal{O}{X,x}$. Thanks for pointing that out, I’ve can show that they’re isomorphic as rings, but I think I’ve been imprecise in stating my isomorphism of modules, so I’ll edit my question to make things more explicit – Dave Jul 03 '19 at 13:33
  • I’ve also added my thoughts having spent more time on this – Dave Jul 03 '19 at 14:02
  • Essentially you should prove that if you start with a ring $R$, an $R$-module $M$ and a prime ideal $\mathfrak{p}$ of $R$ that $M_{\mathfrak{p}}$ is (canonically) isomorphic to $\mathcal{F}x$ as an $R{\mathfrak{p}}$ module. Here $\mathcal{F}$ is the quasi-coherent sheaf on $X=\spec R$ associated to $M$ and $x \in X$ is the point associated to $\mathfrak{p}$. Passing to smaller and smaller sets can be made precise by talking about taking colimits over `cofinal' indexing sets. – Pol van Hoften Jul 03 '19 at 17:24
  • @user45878 Many thanks for your reply, I think that I can show this, but the issue I'm having is that my $X$ is not necessarily an affine variety/scheme, and even if it were, I'd have two isomorphic abelian groups acted on separately by two isomorphic rings, but the actions might be very different so the number of generators could change – Dave Jul 03 '19 at 17:48
  • You can always reduce to the case that $X$ is affine (the set of affine open neighborhoods of $x$ is cofinal in the set of all open neighborhoods). Secondly, the isomorphism you get is an isomorphism of $R_{\mathfrak{P}}$-modules (not just of abelian groups). – Pol van Hoften Jul 03 '19 at 18:09
  • @user45878 Thank you, I think I'm close to understanding this now, I've updated my question with what I think would show the result along these lines? – Dave Jul 03 '19 at 22:11

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