Descending etale morphisms

Helper section to prove descent for qc + sep etale morphisms
relative to surjective integral morphisms

etale.tex

@@ -1976,6 +1976,366 @@ \section{Permanence properties}
 
 
 
+\section{Descending \'etale morphisms}
+\label{section-descending-etale}
+
+\noindent
+In order to understand the language used in this section we encourage
+the reader to take a look at
+Descent, Section \ref{descent-section-descent-datum}.
+Let $f : X \to S$ be a morphism of schemes. Consider the
+pullback functor
+\begin{equation}
+\label{equation-descent-etale}
+\text{schemes }U\text{ \'etale over }S \longrightarrow
+\begin{matrix}
+\text{descent data }(V, \varphi)\text{ relative to }X/S \\
+\text{ with }V\text{ \'etale over }X
+\end{matrix}
+\end{equation}
+sending $U$ to the canonical descent datum $(X \times_S U, can)$.
+
+\begin{lemma}
+\label{lemma-faithful}
+If $f : X \to S$ is surjective, then the functor
+(\ref{equation-descent-etale}) is faithful.
+\end{lemma}
+
+\begin{proof}
+Let $a, b : U_1 \to U_2$ be two morphisms between schemes \'etale over $S$.
+Assume the base changes of $a$ and $b$ to $X$ agree.
+We have to show that $a = b$.
+By Proposition \ref{proposition-equality} it suffices to
+show that $a$ and $b$ agree on points and residue fields.
+This is clear because for every $u \in U_1$ we can find a point
+$v \in X \times_S U_1$ mapping to $u$.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-fully-faithful}
+Assume $f : X \to S$ is submersive and any \'etale base change
+of $f$ is submersive. Then the functor
+(\ref{equation-descent-etale}) is fully faithful.
+\end{lemma}
+
+\begin{proof}
+By Lemma \ref{lemma-faithful} the functor is faithful.
+Let $U_1 \to S$ and $U_2 \to S$ be \'etale morphisms
+and let $a : X \times_S U_1 \to X \times_S U_2$ be a
+morphism compatible with canonical descent data.
+We will prove that $a$ is the base change of a morphism $U_1 \to U_2$.
+
+\medskip\noindent
+Let $U'_2 \subset U_2$ be an open subscheme. Consider
+$W = a^{-1}(X \times_S U'_2)$. This is an open subscheme
+of $X \times_S U_1$ which is compatible with the canonical
+descent datum on $V_1 = X \times_S U_1$. This means that the
+two inverse images of $W$ by the projections
+$V_1 \times_{U_1} V_1 \to V_1$ agree. Since $V_1 \to U_1$
+is surjective (as the base change of $X \to S$) we conclude
+that $W$ is the inverse image of some subset $U'_1 \subset U_1$.
+Since $W$ is open, our assumption on $f$ implies that $U'_1 \subset U_1$
+is open.
+
+\medskip\noindent
+Let $U_2 = \bigcup U_{2, i}$ be an affine open covering.
+By the result of the preceding paragraph we obtain an open
+covering $U_1 = \bigcup U_{1, i}$ such that
+$X \times_S U_{1, i} = a^{-1}(X \times_S U_{2, i})$.
+If we can prove there exists a morphism $U_{1, i} \to U_{2, i}$
+whose base change is the morphism
+$a_i : X \times_S U_{1, i} \to X \times_S U_{2, i}$
+then we can glue these morphisms to a morphism $U_1 \to U_2$
+(using faithfulness). In this way we reduce to the case that
+$U_2$ is affine. In particular $U_2 \to X$ is separated
+(Schemes, Lemma \ref{schemes-lemma-compose-after-separated}).
+
+\medskip\noindent
+Assume $U_2 \to S$ is separated. Then the graph $\Gamma_a$ of $a$
+is a closed subscheme of
+$$
+V = (X \times_S U_1) \times_X (X \times_S U_2) = X \times_S U_1 \times_S U_2
+$$
+by Schemes, Lemma \ref{schemes-lemma-semi-diagonal}.
+On the other hand the graph is open for example
+because it is a section of an \'etale morphism
+(Proposition \ref{proposition-properties-sections}).
+Since $a$ is a morphism of descent data, the two inverse images of
+$\Gamma_a \subset V$ under the projections
+$V \times_{U_1 \times_S U_2} V \to V$ are the same.
+Hence arguing as in the second paragraph of the proof we
+find an open and closed subscheme $\Gamma \subset U_1 \times_S U_2$
+whose base change to $X$ gives $\Gamma_a$. Then
+$\Gamma \to U_1$ is an \'etale morphism whose base change
+to $X$ is an isomorphism. This means that $\Gamma \to U_1$
+is universally bijective, hence an isomorphism
+by Theorem \ref{theorem-etale-radicial-open}.
+Thus $\Gamma$ is the graph of a morphism $U_1 \to U_2$
+and the base change of this morphism is $a$ as desired.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-fully-faithful-cases}
+Let $f : X \to S$ be a morphism of schemes. In the following
+cases the functor (\ref{equation-descent-etale}) is fully faithful:
+\begin{enumerate}
+\item $f$ is surjective and universally closed
+(e.g., finite, integral, or proper),
+\item $f$ is surjective and universally open
+(e.g., locally of finite presentation and flat, smooth, or etale),
+\item $f$ is surjective, quasi-compact, and flat.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+This follows from Lemma \ref{lemma-fully-faithful}.
+For example a closed surjective map of topological spaces
+is submersive (Topology, Lemma
+\ref{topology-lemma-closed-morphism-quotient-topology}).
+Finite, integral, and proper morphisms are universally closed, see
+Morphisms, Lemmas \ref{morphisms-lemma-integral-universally-closed} and
+\ref{morphisms-lemma-finite-proper} and
+Definition \ref{morphisms-definition-proper}.
+On the other hand an open surjective map of topoological spaces
+is submersive (Topology, Lemma
+\ref{topology-lemma-open-morphism-quotient-topology}).
+Flat locally finitely presented, smooth, and \'etale morphisms are
+universally open, see
+Morphisms, Lemmas \ref{morphisms-lemma-fppf-open},
+\ref{morphisms-lemma-smooth-open}, and
+\ref{morphisms-lemma-etale-open}.
+The case of surjective, quasi-compact, flat morphisms follows
+from Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology}.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-reduce-to-affine}
+Let $f : X \to S$ be a morphism of schemes.
+Let $(V, \varphi)$ be a descent datum relative to $X/S$
+with $V \to X$ \'etale. Let $S = \bigcup S_i$ be an
+open covering. Assume that
+\begin{enumerate}
+\item the pullback of the descent datum $(V, \varphi)$
+to $X \times_S S_i/S_i$ is effective,
+\item the functor (\ref{equation-descent-etale})
+for $X \times_S (S_i \cap S_j) \to (S_i \cap S_j)$ is fully faithful, and
+\item the functor (\ref{equation-descent-etale})
+for $X \times_S (S_i \cap S_j \cap S_k) \to (S_i \cap S_j \cap S_k)$
+is faithful.
+\end{enumerate}
+Then $(V, \varphi)$ is effective.
+\end{lemma}
+
+\begin{proof}
+(Recall that pullbacks of descent data are defined in
+Descent, Definition \ref{descent-definition-pullback-functor}.)
+Set $X_i = X \times_S S_i$. Denote $(V_i, \varphi_i)$ the pullback
+of $(V, \varphi)$ to $X_i/S_i$.
+By assumption (1) we can find an \'etale morphism $U_i \to S_i$
+which comes with an isomorphism $X_i \times_{S_i} U_i \to V_i$ compatible with
+$can$ and $\varphi_i$. By assumption (2) we obtain isomorphisms
+$\psi_{ij} : U_i \times_{S_i} (S_i \cap S_j) \to
+U_j \times_{S_j} (S_i \cap S_j)$.
+By assumption (3) these isomorphisms satisfy the cocycle condition
+so that $(U_i, \psi_{ij})$ is a descend datum for the
+Zariski covering $\{S_i \to S\}$. Then Descent, Lemma
+\ref{descent-lemma-Zariski-refinement-coverings-equivalence}
+(which is essentially just a reformulation of
+Schemes, Section \ref{schemes-section-glueing-schemes})
+tells us that there exists a morphism of schemes $U \to S$
+and isomorphisms $U \times_S S_i \to U_i$ compatible
+with $\psi_{ij}$. The isomorphisms $U \times_S S_i \to U_i$
+determine corresponding isomorphisms $X_i \times_S U \to V_i$
+which glue to a morphism $X \times_S U \to V$ compatible
+with the canonical descent datum and $\varphi$.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-split-henselian}
+Let $(A, I)$ be a henselian pair. Let $U \to \Spec(A)$ be a
+quasi-compact, separated, \'etale morphism such that
+$U \times_{\Spec(A)} \Spec(A/I) \to \Spec(A/I)$ is finite.
+Then
+$$
+U = U_{fin} \amalg U_{away}
+$$
+where $U_{fin} \to \Spec(A)$ is finite and $U_{away}$ has
+no points lying over $Z$.
+\end{lemma}
+
+\begin{proof}
+By Zariski's main theorem, the scheme $U$ is quasi-affine.
+In fact, we can find an open immersion $U \to T$ with $T$ affine and
+$T \to \Spec(A)$ finite, see More on Morphisms, Lemma
+\ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite}.
+Write $Z = \Spec(A/I)$ and denote $U_Z \to T_Z$ the base change.
+Since $U_Z \to Z$ is finite, we see that $U_Z \to T_Z$ is closed
+as well as open. Hence by
+More on Algebra, Lemma \ref{more-algebra-lemma-characterize-henselian-pair}
+we obtain a unique decomposition $T = T' \amalg T''$ with $T'_Z = U_Z$.
+Set $U_{fin} = U \cap T'$ and $U_{away} = U \cap T''$. Since
+$T'_Z \subset U_Z$ we see that all closed points of $T'$ are in $U$
+hence $T' \subset U$, hence $U_{fin} = T'$, hence $U_{fin} \to \Spec(A)$
+is finite. We omit the proof
+of uniqueness of the decomposition.
+\end{proof}
+
+\begin{proposition}
+\label{proposition-effective}
+Let $f : X \to S$ be a surjective integral morphism.
+Let $(V, \varphi)$ be a descent datum relative to $X/S$
+with $V \to X$ quasi-compact, separated, and \'etale.
+Then $(V, \varphi)$ is effective.
+\end{proposition}
+
+\begin{proof}
+By Lemma \ref{lemma-fully-faithful-cases} the
+functor (\ref{equation-descent-etale})
+is fully faithful and the same remains the case after any
+base change $S \to S'$. Hence we can use
+Lemma \ref{lemma-reduce-to-affine}
+to see that it suffices to prove the effectivity
+Zariski locally on $S$. In particular we may and do
+assume that $S$ is affine.
+
+\medskip\noindent
+If $S$ is affine we can find a directed partially ordered
+set $\Lambda$ and an inverse system $X_\lambda \to S_\lambda$
+of finite morphisms of affine schemes of finite type over
+$\Spec(\mathbf{Z})$ such that $(X \to S) = \lim (X_\lambda \to S_\lambda)$.
+See Algebra, Lemma \ref{algebra-lemma-limit-integral}.
+Since limits commute with limits we deduce that
+$X \times_S X = \lim X_\lambda \times_{S_\lambda} X_\lambda$
+and 
+$X \times_S X \times_S X = \lim
+X_\lambda \times_{S_\lambda} X_\lambda \times_{S_\lambda} X_\lambda$.
+Observe that $V \to X$ is a morphism of finite presentation.
+Using Limits, Lemmas \ref{limits-lemma-descend-finite-presentation}
+we can find an $\lambda$ and a descent datum $(V_\lambda, \varphi_\lambda)$
+relative to $X_\lambda/S_\lambda$ whose pullback to $X/S$ is
+$(V, \varphi)$. Of course it is enough to show that
+$(V_\lambda, \varphi_\lambda)$ is effective. Note that $V_\lambda$
+is quasi-compact by construction.
+After possibly increasing $\lambda$ we may assume
+that $V_\lambda \to X_\lambda$ is separated and \'etale, see
+Limits, Lemma \ref{limits-lemma-descend-separated-finite-presentation} and
+\ref{limits-lemma-descend-etale}.
+Thus we may assume that $f$ is finite surjective and
+$S$ affine of finite type over $\mathbf{Z}$.
+
+\medskip\noindent
+Consider an open $S' \subset S$ such that the pullback $(V', \varphi')$
+of $(V, \varphi)$ to $X' = X \times_S S'$ is effective. Below we will
+prove, that $S' \not = S$ implies there is a strictly larger open over
+which the descent datum is effective. Since $S$ is Noetherian (and hence
+has a Noetherian underlying topological space) this will finish the proof.
+Let $\xi \in S$ be a generic point of an irreducible component of the
+closed subset $Z = S \setminus S'$.
+If $\xi \in S'' \subset S$ is an open over which the descent datum is
+effective, then the descent datum is effective over
+$S' \cup S''$ by the glueing argument of the first paragraph. Thus
+in the rest of the proof we may replace $S$ by an affine open
+neighbourhood of $\xi$.
+
+\medskip\noindent
+After a first such replacement we may assume that $Z$ is irreducible
+with generic point $Z$. Let us endow $Z$ with the reduced induced
+closed subscheme structure. After another shrinking we may assume
+$X_Z = X \times_S Z = f^{-1}(Z) \to Z$ is flat, see
+Morphisms, Proposition \ref{morphisms-proposition-generic-flatness}.
+Let $(V_Z, \varphi_Z)$ be the pullback of the descent datum to $X_Z/Z$.
+By More on Morphisms, Lemma
+\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}
+this descent datum is effective and we obtain an \'etale morphism
+$U_Z \to Z$ whose base change is isomorphic to $V_Z$ in a manner
+compatible with descent data.
+Of course $U_Z \to Z$ is quasi-compact and separated
+(Descent, Lemmas \ref{descent-lemma-descending-property-quasi-compact} and
+\ref{descent-lemma-descending-property-separated}).
+Thus after shrinking once more we may assume
+that $U_Z \to Z$ is finite, see
+Morphisms, Lemma \ref{morphisms-lemma-generically-finite}.
+
+\medskip\noindent
+Let $S = \Spec(A)$ and let $I \subset A$ be the prime ideal corresponding
+to $Z \subset S$. Let $(A^h, IA^h)$ be the henselization of the pair
+$(A, I)$. Denote $S^h = \Spec(A^h)$ and $Z^h = V(IA^h) \cong Z$.
+We claim that it suffices to show effectivity after base change to
+$S^h$. Namely, $\{S^h \to S, S' \to S\}$ is an fpqc covering
+($A \to A^h$ is flat by More on Algebra, Lemma
+\ref{more-algebra-lemma-henselization-flat}) and
+by More on Morphisms, Lemma
+\ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend}
+we have fpqc descent for separated \'etale morphisms.
+Namely, if $U^h \to S^h$ and $U' \to S'$ are the objects
+corresponding to the pullbacks $(V^h, \varphi^h)$ and
+$(V', \varphi')$, then the required isomorphisms
+$$
+U^h \times_S S^h \to S^h \times_S V^h
+\quad\text{and}\quad
+U^h \times_S S' \to S^h \times_S U'
+$$
+are obtained by the fully faithfulness pointed out in the first
+paragraph. In this way we reduce to the situation described in
+the next paragraph.
+
+\medskip\noindent
+Here $S = \Spec(A)$, $Z = V(I)$, $S' = S \setminus Z$ where
+$(A, I)$ is a henselian pair, we have $U' \to S'$ corresponding
+to the descent datum $(V', \varphi')$ and we have a finite \'etale
+morphism $U_Z \to Z$ corresponding to the descent datum
+$(V_Z, \varphi_Z)$. We no longer have that $A$ is of finite type
+over $\mathbf{Z}$; but the rest of the argument will not even use
+that $A$ is Noetherian.
+By More on Algebra, Lemma \ref{more-algebra-lemma-finite-etale-equivalence}
+we can find a finite \'etale morphism $U_{fin} \to S$ whose
+restriction to $Z$ is isomorphic to $U_Z \to Z$.
+Write $X = \Spec(B)$ and $Y = V(IB)$. Since $(B, IB)$ is a henselian pair
+(More on Algebra, Lemma \ref{more-algebra-lemma-integral-over-henselian-pair})
+and since the restriction $V \to X$ to $Y$
+is finite (as base change of $U_Z \to Z$) we see that
+there is a canonical disjoint union decomposition
+$$
+V = V_{fin} \amalg V_{away}
+$$
+were $V_{fin} \to X$ is finite and where $V_{away}$ has no
+points lying over $Y$. See Lemma \ref{lemma-split-henselian}.
+Using the uniquenss of this decomposition over $X \times_S X$
+we see that $\varphi$ preserves it and we obtain
+$$
+(V, \varphi) = (V_{fin}, \varphi_{fin}) \amalg (V_{away}, \varphi_{away})
+$$
+in the category of descent data.
+By More on Algebra, Lemma \ref{more-algebra-lemma-finite-etale-equivalence}
+there is a unique isomorphism
+$$
+X \times_S U_{fin} \longrightarrow V_{fin}
+$$
+compatible with the given isomorphism $Y \times_Z U_Z \to V \times_X Y$
+over $Y$.
+By the uniqueness we see that this isomorphism is compatible
+with descent data, i.e.,
+$(X \times_S U_{fin}, can) \cong (V_{fin}, \varphi_{fin})$.
+Denote $U'_{fin} = U_{fin} \times_S S'$. By fully faithfulness
+we obtain a morphism $U'_{fin} \to U'$ which is
+the inclusion of an open (and closed) subscheme.
+Then we set $U = U_{fin} \amalg_{U'_{fin}} U'$ (glueing of schemes as
+in Schemes, Section \ref{schemes-section-glueing-schemes}).
+The morphisms $X \times_S U_{fin} \to V$ and
+$X \times_S U' \to V$ glue to a morphism $X \times_S U \to V$
+which is the desired isomorphism.
+\end{proof}
+
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 \input{chapters}