## 34.13 Extending functors

Let us start with a simple example which explains what we are doing. Let $R$ be a ring. Suppose $F$ is a functor defined on the category $\mathcal{C}$ of $R$-algebras of the form

$A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$

for $n, m \geq 0$ integers and $f_1, \ldots , f_ m \in R[x_1, \ldots , x_ n]$ elements. Then for any $R$-algebra $B$ we can define

$F'(B) = \mathop{\mathrm{colim}}\nolimits _{A \to B,\ A \in \mathcal{C}} F(A)$

It turns out $F'$ is the unique functor on the category of all $R$-algebras which extends $F$ and commutes with filtered colimits. The same procedure works in the category of schemes if we impose that our functor is a Zariski sheaf.

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. $\square$

Lemma 34.13.2. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\}$. Let $T$ be an affine scheme which is written as a limit $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ of a directed inverse system of affine schemes.

1. Let $\mathcal{V} = \{ V_ j \to T\} _{j = 1, \ldots , m}$ be a standard $\tau$-covering of $T$, see Definitions 34.3.4, 34.4.5, 34.5.5, 34.6.5, and 34.7.5. Then there exists an index $i$ and a standard $\tau$-covering $\mathcal{V}_ i = \{ V_{i, j} \to T_ i\} _{j = 1, \ldots , m}$ whose base change $T \times _{T_ i} \mathcal{V}_ i$ to $T$ is isomorphic to $\mathcal{V}$.

2. Let $\mathcal{V}_ i$, $\mathcal{V}'_ i$ be a pair of standard $\tau$-coverings of $T_ i$. If $f : T \times _{T_ i} \mathcal{V}_ i \to T \times _{T_ i} \mathcal{V}'_ i$ is a morphism of coverings of $T$, then there exists an index $i' \geq i$ and a morphism $f_{i'} : T_{i'} \times _{T_ i} \mathcal{V} \to T_{i'} \times _{T_ i} \mathcal{V}'_ i$ whose base change to $T$ is $f$.

3. If $f, g : \mathcal{V} \to \mathcal{V}'_ i$ are morphisms of standard $\tau$-coverings of $T_ i$ whose base changes $f_ T, g_ T$ to $T$ are equal then there exists an index $i' \geq i$ such that $f_{T_{i'}} = g_{T_{i'}}$.

In other words, the category of standard $\tau$-coverings of $T$ is the colimit over $I$ of the categories of standard $\tau$-coverings of $T_ i$.

Proof. Let us prove this for $\tau = fppf$. By Limits, Lemma 32.10.1 the category of schemes of finite presentation over $T$ is the colimit over $I$ of the categories of finite presentation over $T_ i$. By Limits, Lemmas 32.8.2 and 32.8.7 the same is true for category of schemes which are affine, flat and of finite presentation over $T$. To finish the proof of the lemma it suffices to show that if $\{ V_{j, i} \to T_ i\} _{j = 1, \ldots , m}$ is a finite family of flat finitely presented morphisms with $V_{j, i}$ affine, and the base change $\coprod _ j T \times _{T_ i} V_{j, i} \to T$ is surjective, then for some $i' \geq i$ the morphism $\coprod T_{i'} \times _{T_ i} V_{j, i} \to T_{i'}$ is surjective. Denote $W_{i'} \subset T_{i'}$, resp. $W \subset T$ the image. Of course $W = T$ by assumption. Since the morphisms are flat and of finite presentation we see that $W_ i$ is a quasi-compact open of $T_ i$, see Morphisms, Lemma 29.25.10. Moreover, $W = T \times _{T_ i} W_ i$ (formation of image commutes with base change). Hence by Limits, Lemma 32.4.11 we conclude that $W_{i'} = T_{i'}$ for some large enough $i'$ and we win.

For $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic\}$ a standard $\tau$-covering is a standard fppf covering. Hence the fully faithfulness of the functor holds. The only issue is to show that given a standard fppf covering $\mathcal{V}_ i$ for some $i$ such that $\mathcal{V}_ i \times _{T_ i} T$ is a standard $\tau$-covering, then $\mathcal{V}_ i \times _{T_ i} T_{i'}$ is a standard $\tau$-covering for all $i' \gg i$. This follows immediately from Limits, Lemmas 32.8.12, 32.8.10, 32.8.9, and 32.8.16. $\square$

Lemma 34.13.3. Let $S$, $\mathcal{C}$, $F$ satisfy conditions (1), (2), (a), and (b) of Lemma 34.13.1 and denote $F' : (\mathit{Sch}/S)^{opp} \to \textit{Sets}$ the unique extension constructed in the lemma. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\}$. Assume

1. for any standard $\tau$-covering $\{ V_ i \to V\} _{i = 1, \ldots , n}$ of affines in $\mathit{Sch}/S$ such that $V \to S$ factors through an affine open $U \subset S$ and $V \to U$ is of finite presentation, the sheaf condition hold for $F$ and $\{ V_ i \to V\} _{i = 1, \ldots , n}$2.

Then $F'$ satisfies the sheaf condition for all $\tau$-coverings.

Proof. Let $X$ be a scheme over $S$ and let $\{ X_ i \to X\} _{i \in I}$ be a $\tau$-covering. Let $s_ i \in F'(X_ i)$ be elements such that $s_ i$ and $s_ j$ map to the same element of $F'(X_ i \times _ X X_ j)$ for all $i, j \in I$. We have to show that there is a unique element $s \in F'(X)$ restricting to $s_ i \in F'(X_ i)$ for all $i \in I$.

Special case: $X$ is an affine such that the structure morphism maps into an affine open $U$ of $S$ and the covering $\{ X_ i \to X\} _{i \in I}$ is a standard $\tau$-covering. In this case we can write

$X = \mathop{\mathrm{lim}}\nolimits V_ k$

as a cofiltered limit with $V_ k \to U$ of finite presentation and $V_ k$ affine. See Algebra, Lemma 10.127.2. By Lemma 34.13.2 there exists a $k$ and a standard $\tau$-covering $\{ V_{k, i} \to V_ k\} _{i \in I}$ whose base change to $X$ is the given covering. For $k' \geq k$ denote $\{ V_{k', i} \to V_{k'}\} _{i \in I}$ the base change to $V_{k'}$ of our covering. Then we see that

\begin{align*} F'(X) & = \mathop{\mathrm{colim}}\nolimits _{k' \geq k} F(V_ k) \\ & = \mathop{\mathrm{colim}}\nolimits _{k' \geq k} \text{Equalizer}( \xymatrix{ \prod F(V_{k', i}) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod F(V_{k', i} \times _{V_{k'}} V_{k', j}) } \\ & = \text{Equalizer}( \xymatrix{ \mathop{\mathrm{colim}}\nolimits _{k' \geq k} \prod F(V_{k', i}) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathop{\mathrm{colim}}\nolimits _{k' \geq k} \prod F(V_{k', i} \times _{V_{k'}} V_{k', j}) } \\ & = \text{Equalizer}( \xymatrix{ \prod F'(X_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod F'(X_ i \times _ X X_ j) } \end{align*}

The first equality holds by construction of $F'$. The second holds by assumption (c). The third holds because filtered colimits are exact. The fourth again holds by construction of $F'$. In this way we find that the sheaf property holds for $F'$ with respect to $\{ X_ i \to X\} _{i \in I}$.

General case. Choose an affine open covering $X = \bigcup U_ k$ such that each $U_ k$ maps into an affine open of $S$. For every $k$ we may choose a standard $\tau$-covering $\{ V_{k, j} \to U_ k\} _{j = 1, \ldots , m_ k}$ which refines $\{ X_ i \times _ X U_ k \to U_ k\} _{i \in I}$. For each $j \in \{ 1, \ldots , m_ k\}$ choose an index $i_{k, j} \in I$ and a morphism $g_{k, j} : V_{k, j} \to X_{i_{k, j}}$ over $X$. Let $s_{k, j}$ be the element of $F'(V_{k, j})$ we get by restricting $s_{i_{k, j}}$ via $g_{k, j}$. Observe that $s_{k, j}$ and $s_{k', j'}$ restrict to the same element of $F'(V_{k, j} \times _ X V_{k', j'})$ for all $k$ and $k'$ and all $j \in \{ 1, \ldots , m_ k\}$ and $j' \in \{ 1, \ldots , m_{k'}\}$; verification omitted. In particular, by the result of the previous paragraph there is a unique element $s_ k \in F'(U_ k)$ restricting to $s_{k, j}$ for all $j$. With this notation we are ready to finish the proof.

Proof of uniqueness of $s$: this is true because $F'$ satisfies the sheaf property for Zariski coverings and $s|_{U_ k}$ must be equal to $s_ k$ because both restrict to $s_{k, j}$ for all $j$. This uniqueness then shows that $s_ k$ and $s_{k'}$ must restrict to the same section of $F'$ over (the non-affine scheme) $U_ k \cap U_{k'}$ because these sections restrict to the same section over the $\tau$-covering $\{ V_{k, j} \times _ X V_{k', j'} \to U_ k \cap U_{k'}\}$. Thus by the sheaf property for Zariski coverings, there is a unique section $s$ of $F'$ over $X$ whose restriction to $U_ k$ is $s_ k$. We omit the verification (similar to the above) that $s$ restricts to $s_ i$ over $X_ i$. $\square$

Lemma 34.13.4. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\}$. Let $S$ be a scheme contained in a big site $\mathit{Sch}_\tau$. Let $F : (\mathit{Sch}/S)_\tau ^{opp} \to \textit{Sets}$ be a $\tau$-sheaf satisfying property (b) of Lemma 34.13.1 with $\mathcal{C} = (\mathit{Sch}/S)_\tau$. Then the extension $F'$ of $F$ to the category of all schemes over $S$ satisfies the sheaf condition for all $\tau$-coverings.

Proof. This follows from Lemma 34.13.3 applied with $\mathcal{C} = (\mathit{Sch}/S)_\tau$. Conditions (1), (2), (a), and (b) of Lemma 34.13.1 hold; we omit the details. Thus we get our unique extension $F'$ to the category of all schemes over $S$. Finally, observe that any standard $\tau$-covering is tautologically equivalent to a covering in $(\mathit{Sch}/S)_\tau$, see Sets, Lemma 3.9.9 as well as Lemmas 34.3.6, 34.4.7, 34.5.7, 34.6.7, and 34.7.7. By Sites, Lemma 7.8.4 the sheaf property passes through tautological equivalence of coverings. Hence the fact that $F$ is a $\tau$-sheaf implies that property (c) of Lemma 34.13.3 holds and we conclude. $\square$

[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 $\prod F(U_{ijk})$.
[2] This makes sense as $V$, $V_ i$, and $V_ i \times _ V V_ j$ are isomorphic to objects of $\mathcal{C}$ by (2).

Comment #8840 by Yunchi Zheng on

In the first example, is there a typo that $f_1 ,..., f_m \in R[x_1,..., x_m]$ should be $f_1 ,..., f_m \in R[x_1,..., x_n]$?

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