Is it known whether or not every sheaf of ideals of the etale structure sheaf of a Noetherian scheme is generated by finitely many of its sections? Of course it is trivially true for some widely used special cases. But is it known one way or the other, in this generality?
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2$\begingroup$ what is the etale structure sheaf? $\endgroup$– Yosemite SamCommented Jul 17, 2012 at 16:00
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1$\begingroup$ The sheaf $U \mapsto \mathcal{O}_X(U)$ defined on étale opens $U \to X$. $\endgroup$– Martin BrandenburgCommented Jul 17, 2012 at 18:01
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1$\begingroup$ Yes, though I would write $U\mapsto \mathscr{O}_U(U)$ to define it. $\endgroup$– Colin McLartyCommented Jul 17, 2012 at 19:17
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1$\begingroup$ OK guys, I'm being really really thick right now. What's the difference with the Zariski structure sheaf? Aren't the two categories (etale/zariski quasi-coherent modules) equivalent for schemes (via the forget morphism)? $\endgroup$– Yosemite SamCommented Jul 17, 2012 at 22:44
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$\begingroup$ But I need not only quasi-coherent (sheaves of) ideals. I want to know this for all (sheaves of) ideals. $\endgroup$– Colin McLartyCommented Jul 17, 2012 at 23:11
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1 Answer
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I must apologize for posting a false answer. in writing up a proof i discovered a gap which grew to a counterexample.
In fact not every sheaf of ideals of an etale structure sheaf is finitely generated. I have added a counterexample to the end of my ArXiv paper on cohomology in second order arithmetic arXiv:1207.0276v2. Intuitively, an etale ideal can hold information about arbitrarily high degree extensions which is not reducible to information about any fixed degree so the ideal is not finitely generated.
The counterexample shows that given any non-zero element $x$ of an algebraically closed field $k$ a single etale sheaf of ideals on the punctuated line $k\0$ can pick one $2^n$-th root of $x$ for every $n$. The example has $x=1$ but rescaling $k\0$ takes 1 to any other $a\neq 0$.