[Functor of Points I] Representable Functors and Zarisky Covering of a Functor

1. Representable Functors.

Let $\mathscr{C}$ be a category, we denote $
\widehat{\mathscr{C}}$ the category of functors from $\mathscr{C}^\text{opp}$ to $Sets$, the Yoneda's lemma tells us there is a faithful functor from $\mathscr{C}$ to $\widehat{\mathscr{C}}$ defined by $X\mapsto h_X$, where $h_X(-):=Hom_{\mathscr{C}}(-,X)$. This implies that we can study $X$ via its associated functor $h_X$. We also recall that a functor $F: \mathscr{C}^\text{opp}\to Sets$ is said to be representable if there exists an object $X$ in $\mathscr{C}$, such that $F\cong h_X$.

If we replace $C$ by the category $Sch$ of schemes, then in general, determining if a functor $F: \widehat{Sch}^\text{opp}\to Sets$ is representable or not is an interesting question. Let us take a look on some examples.

Example 1.1. We define the global section functor $F: Sch^\text{opp}\to Sets$ defined by $X\mapsto \mathscr{O}_X(X)$, then it is representable by $Spec(\mathbb{Z}[t])$. In fact, this follows from the fact that $\mathscr{O}_X(X)\cong Hom_{Rings}(\mathbb{Z}[t], \mathscr{O}_X(X))$, and $Hom_{Rings}(\mathbb{Z}[t], \mathscr{O}_X(X))\cong Hom_{Sch}(X, Spec\mathbb{Z}[t])$ as sets.

Example 1.2. Let $F: Sch^\text{opp}\to Sets$ be a functor defined by $X\mapsto \mathscr{O}_X(X)^\times$, we have $\mathscr{O}_X(X)^\times \cong Hom_{Rings}(\mathbb{Z}[t,t^{-1}], \mathscr{O}_X(X))\cong Hom_{Sch}(X,Spec \mathbb{Z}[t,t^{-1}])$, and we get that $F$ is representable by $Spec \mathbb{Z}[t,t^{-1}]$.

Example 1.3. Let $\mathscr{C}$ be an arbitrary category. We note that in $\widehat{\mathscr{C}}$, the fiber product always exists. Indeed, let $F, G, H$ be functors from $\mathscr C^\text{opp}$ to $Sets$, with $f: F\to H, g:G\to H$ morphisms of functors, we can define

$$(F\times_H G)(T):=F(T)\times_{H(T)}G(T)$$

where the RHS is a fiber product in the category of sets. Then $F\times_H G$ is a well-defined functor from $\mathscr C^\text{opp}$ to $Sets$, and it satisfies the universal properties defining fiber product. Applying this to the situation of schemes, with $H=h_S, F=h_X, G=h_Y$, then we can see that $F
\times_H G$ is representable by $X\times_S Y$. This fact follows easily from the universal property of fiber products.

In the later parts of this series, we will see much more difficult examples, about Grassmanian functors, and Brauer-Severi functors. About $h_X$, we should note that it is deeper than just a functor. We denote $X(T):=h_X(T)$, and we say $X(T)$ is the set of $T$-rational points of $X$. For this notion, I remind to look at my previous notes on $k$-rational points on an affine scheme over fields.

2. Zarisky open covering of a functor.
 
 We are now going to work with schemes basically by their associated functors.

Definition. Let $F,G$ be in $\widehat{Sch}$, a morphism $f: F\to G$ is said to be representable if for all scheme $X$, and all morphism $g: h_X\to G$, the functor $F\times_G h_X$ is representable.

Example 2.1. Let $F:=h_Y, G:=h_Z$, then a morphism from $h_Y\to h_Z$ corresponds to a morphism from $Y\to Z$. In this case, we can form the fiber product $Y\times_Z X$ in the category of schemes, and it represents the functor $h_Y\times_{h_Z}h_X$.

Definition. Let $P$ be a property of morphism of schemes, and $F,G$ in $\widehat{Sch}$. We say that a representable morphism $f: F\to G$ has property $P$ if for all $g: h_X\to G$, and a scheme $Z$ represents $F\times_G h_X$, the corresponding morphism $Z\to X$ has property $P$.

Example 2.2. Let $P=``\text{open immersion}"$, $F:=h_Y, G:=h_Z$, then a morphism $f: h_Y\to h_Z$ corresponds to a morphism $f: Y\to Z$, and it is always representable by the example above. In this case, $h_Y\times_{h_Z}h_X$ is representable by $Y\times_Z X$, and because open immersion is stable under base changes, we can see that the projection map $Y\times_Z X\to X$ is also an open immersion.

We now come to the notion of Zarisky sheaf via functor of points. Let $F: Sch^\text{opp}\to Sets$ be a functor, and $i: U\to X$ is an open immersion of schemes. Let $\psi\in F(X)$, we denote $psi|_{U}:=F(i)(\psi)$.

Definition. Let $F: Sch^\text{opp}\to Sets$ be a functor, we say that $F$ is a Zarisky sheaf if for all scheme $X$, all open covering $\{U_i\}_i$ of $X$, and all $\psi_i\in F(U_i)$ such that for all $i,j$, $\psi_i|_{U_i\cap U_j}=\psi_j|_{U_i\cap U_j}$, there exists a unique $\psi$ in $F(X)$ such that $\psi|_{U_i}=\psi_i$.

Example 2.3. Let $F: Sch^\text{opp}\to Sets$ defined by $F(X):=\mathscr{O}_X(X)$, then by properties of sheaves, $F$ is a Zarisky sheaf.

Example 2.4. Let $F:=h_S$ be a representable functor, then because any morphism from $X$ to $S$ can be glued by morphisms from all $U_i$ to $S$, where $\{U_i\}_i$ is an open covering of $X$, with the condition that these morphism agree on the intersection. We can see that $h_S$ is a Zarisky sheaf.

Definition. Let $F$ be in $\widehat{Sch}$, an open subfunctor of $F$ is an object $F'$ in $\widehat{Sch}$ together with a representable morphism $f: F'\to F$, such that for all $g: h_X\to F$, and $Z$ the scheme representing $F'\times_Fh_X$, the morphism $Z\to X$ is an open immersion.

Example 2.5. Let $F:=h_Z$, and $Y$ is an open subscheme of $Z$, then the corresponding morphism $h_Y\to h_Z$ together with $h_Y$ is an open subfunctor of $h_Z$.

Definition. Let $F$ be a functor, and $\{f_i: F_i\to F\}_i$ be open subfunctors of $F$. We say that $\{f_i:F_i\to F\}$ is an Zarisky open covering of $F$, if for all $g: h_X\to F$, and $Z_i$ is the scheme representing $F_i\times_F h_X$, with the induced map $g_i: Z_i\to X$, then $g_i(Z_i)$ forms an open covering of $X$.

Example 2.6. Let $F:=h_Y$, and $F_i:=h_{U_i}$, where $\{U_i\}_i$ is an open covering of $Y$, then $h_{U_i}$ is an open covering of $h_Y$, since $\{U_i\times_Y X\}_i$ is an open covering of $Y\times_YX=X$.


There is an important theorem, whose proof is more or less the same with the construction of gluing schemes.

Theorem 2.7. Let $F: Sch^\text{opp}\to Sets$ be a functors, then $F$ is representable iff

(i) $F$ is a Zarisky sheaf.

(ii) $F$ has a Zarisky open covering $\{F_i\}_i$, and each $F_i$ is representable.

 
 Application I (Fiber product in the category of schemes). Via the two sections above, we use many examples related to fiber product in the category of schemes. But here is the proof of the existence of fiber product by Grothendieck. First, we can see that there exists fiber product in the category $\widehat{Sch}$ by Example 1.3 and we next use the theorem above to reduce to the affine case. But in the later case, things are done by tensor product.

Later, we will construct Grassmanians over any base scheme, by proving that the Grassmanian functor is representable.