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Suppose I have a scheme $X$ and I want to know if $X$ is reduced, but all I have access to is the functor $$ R\mapsto X(R)=Mor(\operatorname{Spec}(R),X) $$

from commutative rings to sets (rather than, say, an explicit covering by spectra of some rings). Is there some nice criterion for this?

Some thoughts: The universal property of the reduction is that every map from a reduced scheme $Y$ to $X$, factors uniquely through $X_{red}$. Thus, reducedness of $X$ seems to be "in the opposite direction" in the sense that it says something about morphisms from $X$ (that they always factor through the reduction of the target) rather than morphisms to $X$.

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    $\begingroup$ I also asked this myself a while ago. There was also a question on MO discussing quasi-compactness - which also has no obvious functorial description. But already the affine case seems to be hopeless, right? $\endgroup$ Commented Oct 20, 2013 at 20:37
  • $\begingroup$ @MartinBrandenburg, can't you define a space (i.e. sheaf of sets on the category of affine schemes) to be quasi-compact if and only if every covering family has a finite sub-cover? $\endgroup$
    – user314
    Commented Oct 21, 2013 at 15:22
  • $\begingroup$ Sure, but this is just the same definition, probably not useful. Assume you have some moduli problem and prove - using heavy machinery - that there is a moduli space. How do you prove quasi-compactness? $\endgroup$ Commented Oct 21, 2013 at 23:25

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I don't know how to characterize the image of $\mathsf{Sch}_{\mathsf{red}}$ in $\mathsf{Set}^{\mathsf{CRing}}$. But let me mention two things which are related to that question.

  1. Recall that $\mathsf{Sch}$ is equivalent to the full subcategory of $\mathsf{Set}^{\mathsf{CRing}}$ which consists of sheaves in the Zariski topology on $\mathsf{CRing}^{\mathrm{op}}$ (also called local functors in this context) which admit an open covering by representables; the usual reference for this functorial point of view is the book Groupes algébriques by Demazure and Gabriel. Since being reduced is a local property, basically the same proof works with reduced schemes resp. commutative rings. Thus, $\mathsf{Sch}_{\mathrm{red}}$ is equivalent to the full subcategory of $\mathsf{Set}^{\mathsf{CRing}_{\mathsf{red}}}$ which consists of sheaves which have an open covering by representables. I don't see how to extend the embedding $\mathsf{Sch}_{\mathsf{red}} \hookrightarrow \mathsf{Sch}$ to an embedding $\mathsf{Set}^{\mathsf{CRing}_{\mathsf{red}}} \hookrightarrow \mathsf{Set}^{\mathsf{CRing}}$, though. Maybe the left Kan extension works?

  2. The category of reduced commutative rings $\mathsf{CRing}_{\mathsf{red}}$ is a Zariski category in the sense of Yves Diers' book "Categories of commutative algebras", so that we can develope algebraic geometry relative to that. The resulting category of schemes $\mathsf{Sch}_{\mathsf{red}}$ is isomorphic to the usual category of reduced schemes (this follows by some abstract nonsense in Diers' book applied to the forgetful functor $\mathsf{CRing}_{\mathsf{red}} \to \mathsf{CRing}$).

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