Extending functors

Long overdue perhaps... there is a version for stacks too...

topologies.tex

@@ -3967,6 +3967,216 @@ \section{Change of big sites}
 Sites, Lemmas \ref{sites-lemma-when-shriek} and \ref{sites-lemma-bigger-site}.
 
 
+
+
+
+
+\section{Extending functors}
+\label{section-extending-functors}
+
+\noindent
+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 $f_1, \ldots, f_m \in R[x_1, \ldots, x_m]$
+elements. Then for any $R$-algebra $B$ we can define
+$$
+F'(B) = \colim_{A \to B,\ A \in \mathcal{C}} F(A)
+$$
+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.
+
+\begin{lemma}
+\label{lemma-extend}
+Let $S$ be a scheme. Let $\mathcal{C}$ be a full subcategory
+of the category $\Sch/S$ of all schemes over $S$. Assume
+\begin{enumerate}
+\item 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 $\mathcal{C}$,
+\item if $V$ is an affine scheme lying over an affine open $U \subset S$
+such that $V \to U$ is of finite presentation, then $U \to S$ is isomorphic
+to an object of $\mathcal{C}$.
+\end{enumerate}
+Let $F : \mathcal{C}^{opp} \to \textit{Sets}$ be a functor.
+Assume
+\begin{enumerate}
+\item[(a)] 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 family\footnote{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 $\prod F(U_{ijk})$.},
+\item[(b)] if $X = \lim X_i$ is a directed limit of affine schemes
+over $S$ with $X, X_i$ objects of $\mathcal{C}$, then
+$F(X) = \colim F(X_i)$.
+\end{enumerate}
+Then there is a unique way to extend $F$ to a functor
+$(\Sch/S)^{opp} \to \textit{Sets}$ satisfying
+(a) and (b).
+\end{lemma}
+
+\begin{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.
+
+\medskip\noindent
+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
+$$
+V = \lim V_i
+$$
+as a cofiltered limit with $V_i \to U$ of finite presentation
+and $V_i$ affine. See Algebra, Lemma \ref{algebra-lemma-ring-colimit-fp}.
+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
+$$
+F'(V) = \colim F(V_i)
+$$
+Actually, we can give a more canonical expression, namely
+$$
+F'(V) = \colim_{V \to V'} F(V')
+$$
+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
+\ref{limits-proposition-characterize-locally-finite-presentation}
+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).
+
+\medskip\noindent
+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 $V = \lim V_i$ as in
+the previous paragraph. By Limits, Lemma \ref{limits-lemma-descend-opens}
+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
+$$
+F'(V_k) = \colim F(V_{i, k})
+\quad\text{and}\quad
+F'(V_k \cap V_l) = \colim F(V_{i, k} \cap V_{i, l})
+$$
+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.
+
+\medskip\noindent
+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
+\ref{sheaves-lemma-cofinal-systems-coverings-standard-case}
+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 \ref{sheaves-lemma-extend-off-basis}.
+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.
+
+\medskip\noindent
+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 \ref{sheaves-lemma-f-map-basis-above-and-below-structures}.
+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 $F'_X(X)$.
+
+\medskip\noindent
+Property (a) for the functor $X \mapsto F'_X(X)$ is almost immediate
+from the construction; we omit the details.
+Suppose that $X = \lim_{i \in I} X_i$
+is a directed limit of affine schemes over $S$. We have to show that
+$$
+F'_X(X) = \colim_{i \in I} F'_{X_i}(X_i)
+$$
+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
+\ref{limits-proposition-characterize-locally-finite-presentation}.
+Thus we have
+\begin{align*}
+F'_X(X)
+& =
+F'(X)  \\
+& = \colim_{X \to V} F(V) \\
+& =
+\colim_{i' \geq i} \colim_{X_{i'} \to V} F(V) \\
+& =
+\colim_{i' \geq i} F'(X_{i'}) \\
+& =
+\colim_{i' \geq i} F'_{X_{i'}}(X_{i'}) \\
+& =
+\colim_{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
+$$
+F'_{V_k}(V_k) = \colim_{i' \geq i} F'_{V_{i', k}}(V_{i', k})
+$$
+and
+$$
+F'_{V_k \cap V_l}(V_k \cap V_l) =
+\colim_{i' \geq i}
+F'_{V_{i', k} \cap V_{i', l}}(V_{i', k} \cap V_{i', l})
+$$
+By the sheaf property and exactness of filtered colimits
+we find that $F'_X(X) = \colim_{i \in I} F'_{X_i}(X_i)$
+also in this case. This finishes the proof of property (b)
+and hence finishes the proof of the lemma.
+\end{proof}
+
+
+
+
+
+
 \input{chapters}