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Background

Yet another homework inspired question: A scheme is reduced if no section of the structure sheaf is nilpotent. To every scheme $X$ there is a scheme $X_{red}$ and a morphism $i: X_{red} \rightarrow X$ such that every morphism from a reduced scheme into $X$ factors through $X_{red}$. Hartshorne Ex. 2.2.6 guides you through the construction of this scheme. Basically you leave the topological space alone, but you mod out the nilpotents in the structure sheaf. This gives you a presheaf, which you then sheafify to get the structure sheaf of $X_{red}$.

I have been trying to run through all of the constructions in Hartshorne from a functor of points perspective in addition to the "standard" approach, so naturally I was interested in seeing this construction as well.

From this perspective $X$ is a functor $CRing^{op} CRing \rightarrow Sets$, namely $Hom(-,X)$ Hom(Spec(-),X)$if you were using the standard definition of schemes. The functor from$F: CRing \rightarrow CRing$taking$A$to$A/nil(A)$seems relevant here, so it seems natural to ask if$X \circ F^{op}F: CRing^{op} CRing \rightarrow Sets$is the reduced scheme associated to$X$. I won't spell out the details, but this actually turns out to be true (I think at least!). This brings me to my questions. Questions Did I mess up, or does the construction above pan out? Is there a nice characterization of the functors$CRing \rightarrow Cring$which will give a scheme when composed with any scheme$CRing \rightarrow Sets$? Even if there is no simple characterization of all such functors, is there a large class of such functors which is nice? Do you have any other examples of standard constructions in algebraic geometry which are of this form? 1 # Endofunctors of CRing which give schemes when composed with schemes? Background Yet another homework inspired question: A scheme is reduced if no section of the structure sheaf is nilpotent. To every scheme$X$there is a scheme$X_{red}$and a morphism$i: X_{red} \rightarrow X$such that every morphism from a reduced scheme into$X$factors through$X_{red}$. Hartshorne Ex. 2.2.6 guides you through the construction of this scheme. Basically you leave the topological space alone, but you mod out the nilpotents in the structure sheaf. This gives you a presheaf, which you then sheafify to get the structure sheaf of$X_{red}$. I have been trying to run through all of the constructions in Hartshorne from a functor of points perspective in addition to the "standard" approach, so naturally I was interested in seeing this construction as well. From this perspective$X$is a functor$CRing^{op} \rightarrow Sets$, namely$Hom(-,X)$if you were using the standard definition of schemes. The functor from$F: CRing \rightarrow CRing$taking$A$to$A/nil(A)$seems relevant here, so it seems natural to ask if$X \circ F^{op}: CRing^{op} \rightarrow Sets$is the reduced scheme associated to$X$. I won't spell out the details, but this actually turns out to be true (I think at least!). This brings me to my questions. Questions Did I mess up, or does the construction above pan out? Is there a nice characterization of the functors$CRing \rightarrow Cring$which will give a scheme when composed with any scheme$CRing \rightarrow Sets\$?

Even if there is no simple characterization of all such functors, is there a large class of such functors which is nice?

Do you have any other examples of standard constructions in algebraic geometry which are of this form?