The Stacks project

Lemma 34.13.1. Let $S$ be a scheme. Let $\mathcal{C}$ be a full subcategory of the category $\mathit{Sch}/S$ of all schemes over $S$. Assume

  1. if $X \to S$ is an object of $\mathcal{C}$ and $U \subset X$ is an affine open, then $U \to S$ is isomorphic to an object of $\mathcal{C}$,

  2. if $V$ is an affine scheme lying over an affine open $U \subset S$ such that $V \to U$ is of finite presentation, then $V \to S$ is isomorphic to an object of $\mathcal{C}$.

Let $F : \mathcal{C}^{opp} \to \textit{Sets}$ be a functor. Assume

  1. for any Zariski covering $\{ f_ i : X_ i \to X\} _{i \in I}$ with $X, X_ i$ objects of $\mathcal{C}$ we have the sheaf condition for $F$ and this family1,

  2. if $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a directed limit of affine schemes over $S$ with $X, X_ i$ objects of $\mathcal{C}$, then $F(X) = \mathop{\mathrm{colim}}\nolimits F(X_ i)$.

Then there is a unique way to extend $F$ to a functor $F' : (\mathit{Sch}/S)^{opp} \to \textit{Sets}$ satisfying the analogues of (a) and (b), i.e., $F'$ satisfies the sheaf condition for any Zariski covering and $F'(X) = \mathop{\mathrm{colim}}\nolimits F'(X_ i)$ whenever $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a directed limit of affine schemes over $S$.

Proof. The idea will be to first extend $F$ to a sufficiently large collection of affine schemes over $S$ and then use the Zariski sheaf property to extend to all schemes.

Suppose that $V$ is an affine scheme over $S$ whose structure morphism $V \to S$ factors through some affine open $U \subset S$. In this case we can write

\[ V = \mathop{\mathrm{lim}}\nolimits V_ i \]

as a cofiltered limit with $V_ i \to U$ of finite presentation and $V_ i$ affine. See Algebra, Lemma 10.127.2. By conditions (1) and (2) we may replace our $V_ i$ by objects of $\mathcal{C}$. Observe that $V_ i \to S$ is locally of finite presentation (if $S$ is quasi-separated, then these morphisms are actually of finite presentation). Then we set

\[ F'(V) = \mathop{\mathrm{colim}}\nolimits F(V_ i) \]

Actually, we can give a more canonical expression, namely

\[ F'(V) = \mathop{\mathrm{colim}}\nolimits _{V \to V'} F(V') \]

where the colimit is over the category of morphisms $V \to V'$ over $S$ where $V'$ is an object of $\mathcal{C}$ whose structure morphism $V' \to S$ is locally of finite presentation. The reason this is the same as the first formula is that by Limits, Proposition 32.6.1 our inverse system $V_ i$ is cofinal in this category! Finally, note that if $V$ were an object of $\mathcal{C}$, then $F'(V) = F(V)$ by assumption (b).

The second formula turns $F'$ into a contravariant functor on the category formed by affine schemes $V$ over $S$ whose structure morphism factors through an affine open of $S$. Let $V$ be such an affine scheme over $S$ and suppose that $V = \bigcup _{k = 1, \ldots , n} V_ k$ is a finite open covering by affines. Then it makes sense to ask if the sheaf condition holds for $F'$ and this open covering. This is true and easy to show: write $V = \mathop{\mathrm{lim}}\nolimits V_ i$ as in the previous paragraph. By Limits, Lemma 32.4.11 for all sufficiently large $i$ we can find affine opens $V_{i, k} \subset V_ i$ compatible with transition maps pulling back to $V_ k$ in $V$. Thus

\[ F'(V_ k) = \mathop{\mathrm{colim}}\nolimits F(V_{i, k}) \quad \text{and}\quad F'(V_ k \cap V_ l) = \mathop{\mathrm{colim}}\nolimits F(V_{i, k} \cap V_{i, l}) \]

Strictly speaking in these formulas we need to replace $V_{i, k}$ and $V_{i, k} \cap V_{i, l}$ by isomorphic affine objects of $\mathcal{C}$ before applying the functor $F$. Since $I$ is directed the colimits pass through equalizers. Hence the sheaf condition (b) for $F$ and the Zariski coverings $\{ V_{i, k} \to V_ i\} $ implies the sheaf condition for $F'$ and this covering.

Let $X$ be a general scheme over $S$. Let $\mathcal{B}_ X$ denote the collection of affine opens of $X$ whose structure morphism to $S$ maps into an affine open of $S$. It is clear that $\mathcal{B}_ X$ is a basis for the topology of $X$. By the result of the previous paragraph and Sheaves, Lemma 6.30.4 we see that $F'$ is a sheaf on $\mathcal{B}_ X$. Hence $F'$ restricted to $\mathcal{B}_ X$ extends uniquely to a sheaf $F'_ X$ on $X$, see Sheaves, Lemma 6.30.6. If $X$ is an object of $\mathcal{C}$ then we have a canonical identification $F'_ X(X) = F(X)$ by the agreement of $F'$ and $F$ on the objects for which they are both defined and the fact that $F$ satisfies the sheaf condition for Zariski coverings.

Let $f : X \to Y$ be a morphism of schemes over $S$. We get a unique $f$-map from $F'_ Y$ to $F'_ X$ compatible with the maps $F'(V) \to F'(U)$ for all $U \in \mathcal{B}_ X$ and $V \in \mathcal{B}_ Y$ with $f(U) \subset V$, see Sheaves, Lemma 6.30.16. We omit the verification that these maps compose correctly given morphisms $X \to Y \to Z$ of schemes over $S$. We also omit the verification that if $f$ is a morphism of $\mathcal{C}$, then the induced map $F'_ Y(Y) \to F'_ X(X)$ is the same as the map $F(Y) \to F(X)$ via the identifications $F'_ X(X) = F(X)$ and $F'_ Y(Y) = F(Y)$ above. In this way we see that the desired extension of $F$ is the functor which sends $X/S$ to $F'_ X(X)$.

Property (a) for the functor $X \mapsto F'_ X(X)$ is almost immediate from the construction; we omit the details. Suppose that $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ is a directed limit of affine schemes over $S$. We have to show that

\[ F'_ X(X) = \mathop{\mathrm{colim}}\nolimits _{i \in I} F'_{X_ i}(X_ i) \]

First assume that there is some $i \in I$ such that $X_ i \to S$ factors through an affine open $U \subset S$. Then $F'$ is defined on $X$ and on $X_{i'}$ for $i' \geq i$ and we see that $F'_{X_{i'}}(X_{i'}) = F'(X_{i'})$ for $i' \geq i$ and $F'_ X(X) = F'(X)$. In this case every arrow $X \to V$ with $V$ locally of finite presentation over $S$ factors as $X \to X_{i'} \to V$ for some $i' \geq i$, see Limits, Proposition 32.6.1. Thus we have

\begin{align*} F'_ X(X) & = F'(X) \\ & = \mathop{\mathrm{colim}}\nolimits _{X \to V} F(V) \\ & = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \mathop{\mathrm{colim}}\nolimits _{X_{i'} \to V} F(V) \\ & = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} F'(X_{i'}) \\ & = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} F'_{X_{i'}}(X_{i'}) \\ & = \mathop{\mathrm{colim}}\nolimits _{i' \in I} F'_{X_{i'}}(X_{i'}) \end{align*}

as desired. Finally, in general we pick any $i \in I$ and we choose a finite affine open covering $V_ i = V_{i, 1} \cup \ldots \cup V_{i, n}$ such that $V_{i, k} \to S$ factors through an affine open of $S$. Let $V_ k \subset V$ and $V_{i', k}$ for $i' \geq i$ be the inverse images of $V_{i, k}$. By the previous case we see that

\[ F'_{V_ k}(V_ k) = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} F'_{V_{i', k}}(V_{i', k}) \]

and

\[ F'_{V_ k \cap V_ l}(V_ k \cap V_ l) = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} F'_{V_{i', k} \cap V_{i', l}}(V_{i', k} \cap V_{i', l}) \]

By the sheaf property and exactness of filtered colimits we find that $F'_ X(X) = \mathop{\mathrm{colim}}\nolimits _{i \in I} F'_{X_ i}(X_ i)$ also in this case. This finishes the proof of property (b) and hence finishes the proof of the lemma. $\square$

[1] As we do not know that $X_ i \times _ X X_ j$ is in $\mathcal{C}$ this has to be interpreted as follows: by property (1) there exist Zariski coverings $\{ U_{ijk} \to X_ i \times _ X X_ j\} _{k \in K_{ij}}$ with $U_{ijk}$ an object of $\mathcal{C}$. Then the sheaf condition says that $F(X)$ is the equalizer of the two maps from $\prod F(X_ i)$ to $\prod F(U_{ijk})$.

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