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12 votes
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When is a sheaf on a scheme extendable to a representable functor?

If $F$ is a coherent sheaf on a noetherian scheme $X$, there is a natural extension of $F$ to the large Zariski site of $X$: with an object $f\colon T \to X$, you associate the group of global section …
Angelo's user avatar
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15 votes
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Is the functor of open subschemes representable?

No, it is not representable. If it were, the functor of open subset for schemes over a field $k$ would also be representable by the base change to $\mathop{\rm Spec} k$ of the representing object. Sup …
Angelo's user avatar
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4 votes
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When is tensoring with a module representable by a scheme?

When $A$ is noetherian and $M$ is finitely generated, Nitin Nitsure showed that the functor is representable if and only if $M$ is projective (see http://arxiv.org/abs/math/0308036).
Angelo's user avatar
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