Hilbert Schemes II

Next, we introduce the Hilbert scheme functorially following notes by Maclagan. See also Bertram’s course notes, who points out that we (as in Grothendieck) are trying to develop a category-theoretic framework for counting objects with a given property. First we should say a little bit about morphisms of schemes.

To do this, one should indicate that schemes (like manifolds) actually carry the additional structure of a locally ringed space. This means that when you take a kind of (filtered co-)limit on the structure sheaf {\mathcal{O}_X} over all the open neighborhoods containing some point {p\in X}, then we get a local ring {\mathcal{O}_{X,p}}. In this case of an affine scheme {\mathop\mathrm{Spec}{R}} it works out that {\mathcal{O}_{\mathop\mathrm{Spec} R,p}} is actually the localization {R_p}, as {p} is a prime ideal of {R}, and localization satisfies exactly the universal property prescribed by the colimit.

First, remember that schemes are ringed spaces, and that a morphism of ringed spaces {f:X\rightarrow Y} is a continuous map of topological spaces together with a morphism of sheaves from {\mathcal{O}_Y} to the push-forward {f_*\mathcal{O}_X}. This induces a map on stalks

\displaystyle f^\sharp:\mathcal{O}_{Y,q} \rightarrow \mathcal{O}_{X,p}

where {f(p)=q} and supposing {f^\sharp} sends the maximal ideal of one to the other, then we have a morphism of locally ringed spaces, and it is THIS, my friends, which will be a scheme morphism when the locally ringed spaces happen to be schemes… oh hell, this can’t be useful for anyone as exposition; just seriously go skim the first six chapters of The Rising Sea to clear things up. It’s only 200 pages.

Now we have objects and morphisms so we can talk about the category {\mathop\mathrm{Sch}} and functors therefrom. We are interested in the confusingly named functor of points for a given scheme {X}, which is really the one that comes up in many versions of the Yoneda Lemma. The idea is that if category is locally small, meaning the collection of morphisms (arrows) {\mathop\mathrm{Mor}(A,B)} between any two objects forms a set, then we can understand an object by the morphisms going into (or out of) it. Bearing in mind that a contravariant functor between categories {F: \mathcal{C}\rightarrow \mathcal{D}} is the same as a covariant functor {\mathcal{C}^{op}\rightarrow \mathcal{D}}, we have the following.

Definition Suppose {X} is an object of {\mathop\mathrm{Sch}}, the category of schemes. Then there is a contravariant functor {h_X: \mathop\mathrm{Sch}\rightarrow \mathop\mathrm{Set}} such that on objects we have {h_X(Y)=\mathop\mathrm{Mor}(Y,X)}, and for a given morphism {f:Y\rightarrow Z} we have

\displaystyle \begin{array}{rcl} h_X(f):& \mathop\mathrm{Mor}(Z,X) \rightarrow \mathop\mathrm{Mor}(Y,X) \\ & g \mapsto g\circ f \end{array}

In general, functors that come from such {h_X}‘s are said to be representable. Elements of the set {\mathop\mathrm{Mor}(Y,X)} are called the Y-valued points of X, which is weird, because on sight they are just maps. Further, by laziness, in case {Y=\mathop\mathrm{Spec A}} for some ring (or field) A, then we will call theses maps the “{A}-valued points of X.”

Endnote: for an interesting discussion on whether scheme theory can be taught using just {\mathop\mathrm{MaxSpec}}, the maximal ideals of a ring, instead of the primes, that then digresses into thinking of schemes as their functors of points instead of the whole locally ringed space shebang, as well as other AG/Berkeley cultural inside jokes by a bunch of dudes, see the post and extensive comments here. Other interesting background can be found on nLab.

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