Definition 26.15.1. (See Categories, Definition 4.3.6.) Let $F$ be a contravariant functor from the category of schemes to the category of sets (as above). We say that $F$ is *representable by a scheme* or *representable* if there exists a scheme $X$ such that $h_ X \cong F$.

## 26.15 A representability criterion

In this section we reformulate the glueing lemma of Section 26.14 in terms of functors. We recall some of the material from Categories, Section 4.3. Recall that given a scheme $X$ we can define a functor

This is called the *functor of points of $X$*.

Let $F$ be a contravariant functor from the category of schemes to the category of sets. In a formula

We will use the same terminology as in Sites, Section 7.2. Namely, given a scheme $T$, an element $\xi \in F(T)$, and a morphism $f : T' \to T$ we will denote $f^*\xi $ the element $F(f)(\xi )$, and sometimes we will even use the notation $\xi |_{T'}$

Suppose that $F$ is representable by the scheme $X$ and that $s : h_ X \to F$ is an isomorphism. By Categories, Yoneda Lemma 4.3.5 the pair $(X, s : h_ X \to F)$ is unique up to unique isomorphism if it exists. Moreover, the Yoneda lemma says that given any contravariant functor $F$ as above and any scheme $Y$, we have a bijection

Here is the reverse construction. Given any $\xi \in F(Y)$ the transformation of functors $s_\xi : h_ Y \to F$ associates to any morphism $f : T \to Y$ the element $f^*\xi \in F(T)$.

In particular, in the case that $F$ is representable, there exists a scheme $X$ and an element $\xi \in F(X)$ such that the corresponding morphism $h_ X \to F$ is an isomorphism. In this case we also say *the pair $(X, \xi )$ represents $F$*. The element $\xi \in F(X)$ is often called the *“universal family”* for reasons that will become more clear when we talk about algebraic stacks (insert future reference here). For the moment we simply observe that the fact that if the pair $(X, \xi )$ represents $F$, then every element $\xi ' \in F(T)$ for any $T$ is of the form $\xi ' = f^*\xi $ for a unique morphism $f : T \to X$.

Example 26.15.2. Consider the rule which associates to every scheme $T$ the set $F(T) = \Gamma (T, \mathcal{O}_ T)$. We can turn this into a contravariant functor by using for a morphism $f : T' \to T$ the pullback map $f^\sharp : \Gamma (T, \mathcal{O}_ T) \to \Gamma (T', \mathcal{O}_{T'})$. Given a ring $R$ and an element $t \in R$ there exists a unique ring homomorphism $\mathbf{Z}[x] \to R$ which maps $x$ to $t$. Thus, using Lemma 26.6.4, we see that

This does indeed give an isomorphism $h_{\mathop{\mathrm{Spec}}(\mathbf{Z}[x])} \to F$. What is the “universal family” $\xi $? To get it we have to apply the identifications above to $\text{id}_{\mathop{\mathrm{Spec}}(\mathbf{Z}[x])}$. Clearly under the identifications above this gives that $\xi = x \in \Gamma (\mathop{\mathrm{Spec}}(\mathbf{Z}[x]), \mathcal{O}_{\mathop{\mathrm{Spec}}(\mathbf{Z}[x])}) = \mathbf{Z}[x]$ as expected.

Definition 26.15.3. Let $F$ be a contravariant functor on the category of schemes with values in sets.

We say that $F$

*satisfies the sheaf property for the Zariski topology*if for every scheme $T$ and every open covering $T = \bigcup _{i \in I} U_ i$, and for any collection of elements $\xi _ i \in F(U_ i)$ such that $\xi _ i|_{U_ i \cap U_ j} = \xi _ j|_{U_ i \cap U_ j}$ there exists a unique element $\xi \in F(T)$ such that $\xi _ i = \xi |_{U_ i}$ in $F(U_ i)$.A

*subfunctor $H \subset F$*is a rule that associates to every scheme $T$ a subset $H(T) \subset F(T)$ such that the maps $F(f) : F(T) \to F(T')$ maps $H(T)$ into $H(T')$ for all morphisms of schemes $f : T' \to T$.Let $H \subset F$ be a subfunctor. We say that $H \subset F$ is

*representable by open immersions*if for all pairs $(T, \xi )$, where $T$ is a scheme and $\xi \in F(T)$ there exists an open subscheme $U_\xi \subset T$ with the following property:A morphism $f : T' \to T$ factors through $U_\xi $ if and only if $f^*\xi \in H(T')$.

Let $I$ be a set. For each $i \in I$ let $H_ i \subset F$ be a subfunctor. We say that the collection $(H_ i)_{i \in I}$

*covers $F$*if and only if for every $\xi \in F(T)$ there exists an open covering $T = \bigcup U_ i$ such that $\xi |_{U_ i} \in H_ i(U_ i)$.

In condition (4), if $H_ i \subset F$ is representable by open immersions for all $i$, then to check $(H_ i)_{i \in I}$ covers $F$, it suffices to check $F(T) = \bigcup H_ i(T)$ whenever $T$ is the spectrum of a field.

Lemma 26.15.4. Let $F$ be a contravariant functor on the category of schemes with values in the category of sets. Suppose that

$F$ satisfies the sheaf property for the Zariski topology,

there exists a set $I$ and a collection of subfunctors $F_ i \subset F$ such that

each $F_ i$ is representable,

each $F_ i \subset F$ is representable by open immersions, and

the collection $(F_ i)_{i \in I}$ covers $F$.

Then $F$ is representable.

**Proof.**
Let $X_ i$ be a scheme representing $F_ i$ and let $\xi _ i \in F_ i(X_ i) \subset F(X_ i)$ be the “universal family”. Because $F_ j \subset F$ is representable by open immersions, there exists an open $U_{ij} \subset X_ i$ such that $T \to X_ i$ factors through $U_{ij}$ if and only if $\xi _ i|_ T \in F_ j(T)$. In particular $\xi _ i|_{U_{ij}} \in F_ j(U_{ij})$ and therefore we obtain a canonical morphism $\varphi _{ij} : U_{ij} \to X_ j$ such that $\varphi _{ij}^*\xi _ j = \xi _ i|_{U_{ij}}$. By definition of $U_{ji}$ this implies that $\varphi _{ij}$ factors through $U_{ji}$. Since $(\varphi _{ij} \circ \varphi _{ji})^*\xi _ j =\varphi _{ji}^*(\varphi _{ij}^*\xi _ j) = \varphi _{ji}^*\xi _ i = \xi _ j$ we conclude that $\varphi _{ij} \circ \varphi _{ji} = \text{id}_{U_{ji}}$ because the pair $(X_ j, \xi _ j)$ represents $F_ j$. In particular the maps $\varphi _{ij} : U_{ij} \to U_{ji}$ are isomorphisms of schemes. Next we have to show that $\varphi _{ij}^{-1}(U_{ji} \cap U_{jk}) = U_{ij} \cap U_{ik}$. This is true because (a) $U_{ji} \cap U_{jk}$ is the largest open of $U_{ji}$ such that $\xi _ j$ restricts to an element of $F_ k$, (b) $U_{ij} \cap U_{ik}$ is the largest open of $U_{ij}$ such that $\xi _ i$ restricts to an element of $F_ k$, and (c) $\varphi _{ij}^*\xi _ j = \xi _ i$. Moreover, the cocycle condition in Section 26.14 follows because both $\varphi _{jk}|_{U_{ji} \cap U_{jk}} \circ \varphi _{ij}|_{U_{ij} \cap U_{ik}}$ and $\varphi _{ik}|_{U_{ij} \cap U_{ik}}$ pullback $\xi _ k$ to the element $\xi _ i$. Thus we may apply Lemma 26.14.2 to obtain a scheme $X$ with an open covering $X = \bigcup U_ i$ and isomorphisms $\varphi _ i : X_ i \to U_ i$ with properties as in Lemma 26.14.1. Let $\xi _ i' = (\varphi _ i^{-1})^* \xi _ i$. The conditions of Lemma 26.14.1 imply that $\xi _ i'|_{U_ i \cap U_ j} = \xi _ j'|_{U_ i \cap U_ j}$. Therefore, by the condition that $F$ satisfies the sheaf condition in the Zariski topology we see that there exists an element $\xi ' \in F(X)$ such that $\xi _ i = \varphi _ i^*\xi '|_{U_ i}$ for all $i$. Since $\varphi _ i$ is an isomorphism we also get that $(U_ i, \xi '|_{U_ i})$ represents the functor $F_ i$.

We claim that the pair $(X, \xi ')$ represents the functor $F$. To show this, let $T$ be a scheme and let $\xi \in F(T)$. We will construct a unique morphism $g : T \to X$ such that $g^*\xi ' = \xi $. Namely, by the condition that the subfunctors $F_ i$ cover $T$ there exists an open covering $T = \bigcup V_ i$ such that for each $i$ the restriction $\xi |_{V_ i} \in F_ i(V_ i)$. Moreover, since each of the inclusions $F_ i \subset F$ are representable by open immersions we may assume that each $V_ i \subset T$ is maximal open with this property. Because, $(U_ i, \xi '_{U_ i})$ represents the functor $F_ i$ we get a unique morphism $g_ i : V_ i \to U_ i$ such that $g_ i^*\xi '|_{U_ i} = \xi |_{V_ i}$. On the overlaps $V_ i \cap V_ j$ the morphisms $g_ i$ and $g_ j$ agree, for example because they both pull back $\xi '|_{U_ i \cap U_ j} \in F_ i(U_ i \cap U_ j)$ to the same element. Thus the morphisms $g_ i$ glue to a unique morphism from $T \to X$ as desired. $\square$

Remark 26.15.5. Suppose the functor $F$ is defined on all locally ringed spaces, and if conditions of Lemma 26.15.4 are replaced by the following:

$F$ satisfies the sheaf property on the category of locally ringed spaces,

there exists a set $I$ and a collection of subfunctors $F_ i \subset F$ such that

each $F_ i$ is representable by a scheme,

each $F_ i \subset F$ is representable by open immersions on the category of locally ringed spaces, and

the collection $(F_ i)_{i \in I}$ covers $F$ as a functor on the category of locally ringed spaces.

We leave it to the reader to spell this out further. Then the end result is that the functor $F$ is representable in the category of locally ringed spaces and that the representing object is a scheme.

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