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What is an example of a functor $F : \mathbb{C}\text{-Sch.} \to \text{Sets}$ with the property that the restriction of $F$ to locally Noetherian $\mathbb{C}$-schemes can be represented by a locally Noetherian $\mathbb{C}$-scheme, but that scheme does not represent $F$.

I'd be particularly nice to see a "real-world" example (though this might be stretching the notion of "real-world").

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Quot might work--as I recall, the Grothendieck construction restricts to the locally Noetherian category. Off the top of my head, however, I can't prove that Quot isn't representable over $\mathbb{C}$-Sch. – Daniel Litt Nov 2 '10 at 4:32
I suppose an analogy with algebraic topology would be a classifying space BG for bundles on paracompact spaces, but which isn't classifying for all spaces. I'm imagining there might be a moduli problem such that there is a locally Noeth. moduli scheme that is analogous to the alg. top. case. But I'm not an algebraic geometer. – David Roberts Nov 2 '10 at 4:35
Daniel, those are purely expository restrictions. By imposing suitable "finite presentation" conditions that are automatic in the locally noetherian case, all noetherian hypotheses are eliminated (ultimately by reduction to proving that the representing object in noetherian cases actually works on the larger category). – BCnrd Nov 2 '10 at 4:42
up vote 18 down vote accepted

Define $F(X) = {\rm{Hom}}_{\mathbf{C}}(X,{\rm{Spec}}(R/tR))$ where $R$ is the valuation ring of an algebraic closure of $\mathbf{C}((t))$. Note that every element of the maximal ideal of $R/tR$ is nilpotent yet also an $N$th power for arbitrarily large $N$. For any noetherian $\mathbf{C}$-algebra $A$, every $\mathbf{C}$-algebra map $R/tR \rightarrow A$ carries the maximal ideal into the nilradical of $A$. But the nilradical of $A$ has all elements with vanishing $n$th power for some uniform $n$ (depending on $A$) since $A$ is noetherian, so in fact the maximal ideal of $R/tR$ is killed by any such map. In other words, the restriction of $F$ to the full subcategory of locally noetherian objects is represented by ${\rm{Spec}}(\mathbf{C})$.

This is a perfectly "real world" example, since valuation rings of algebraic closures of complete discretely-valued fields come up all the time in number theory and rigid-analytic geometry.

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Thanks! This is indeed a "real-world" example," and much simpler than the examples I was expecting. – jlk Nov 2 '10 at 4:55

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