Permalink
Browse files

Another case of local purity

  • Loading branch information...
aisejohan committed Sep 18, 2018
1 parent bac19bd commit f43c5f2eb7af4dfaa74f9106e72a1eec02498b93
Showing with 343 additions and 5 deletions.
  1. +343 −5 pione.tex
View
348 pione.tex
@@ -4591,8 +4591,9 @@ \section{Purity in local case, I}
fundamental groups of punctured spectra.
These results will be useful to proceed by induction on dimension
in the proofs of our main results on local purity, namely,
Lemma \ref{lemma-local-purity} and
Proposition \ref{proposition-purity-complete-intersection}.
Lemma \ref{lemma-local-purity},
Proposition \ref{proposition-purity-complete-intersection}, and
Proposition \ref{proposition-purity-smooth-over-depth2}.
\begin{lemma}
\label{lemma-sections-over-punctured-spec}
@@ -4932,7 +4933,7 @@ \section{Purity of branch locus}
Set $X = \Spec(A)$ and $U = X \setminus \{\mathfrak m\}$. Then
\begin{enumerate}
\item the functor $\textit{F\'Et}_X \to \textit{F\'Et}_U$
is essentially surjective,
is essentially surjective, i.e., purity holds for $A$,
\item any finite $A \to B$ with $B$ normal which
induces a finite \'etale morphism on punctured spectra is \'etale.
\end{enumerate}
@@ -5075,6 +5076,90 @@ \section{Purity of branch locus}
and the proof is complete.
\end{proof}
\noindent
The following lemma is sometimes useful to find the maximal
open subset over which a finite \'etale morphism extends.
\begin{lemma}
\label{lemma-extend-S2}
Let $j : U \to X$ be an open immersion of locally Noetherian schemes
such that $\text{depth}(\mathcal{O}_{X, x}) \geq 2$ for $x \not \in U$.
Let $\pi : V \to U$ be finite \'etale. Then
\begin{enumerate}
\item $\mathcal{B} = j_*\pi_*\mathcal{O}_V$ is a reflexive coherent
$\mathcal{O}_X$-algebra, set $Y = \underline{\Spec}_X(\mathcal{B})$,
\item $Y \to X$ is the unique finite morphism such that
$V = Y \times_X U$ and $\text{depth}(\mathcal{O}_{Y, y}) \geq 2$
for $y \in Y \setminus V$,
\item $Y \to X$ is \'etale at $y$ if and only if $Y \to X$ is flat at $y$, and
\item $Y \to X$ is \'etale if and only if $\mathcal{B}$
is finite locally free as an $\mathcal{O}_X$-module.
\end{enumerate}
Moreover, the construction of $\mathcal{B}$ and $Y \to X$ commute
with base change by flat morphisms $X' \to X$ of locally Noetherian
schemes.
\end{lemma}
\begin{proof}
Observe that $\pi_*\mathcal{O}_V$ is a finite locally free
$\mathcal{O}_U$-module, in particular reflexive.
By Divisors, Lemma \ref{divisors-lemma-reflexive-S2-extend}
the module $j_*\pi_*\mathcal{O}_V$ is the unique
reflexive coherent module on $X$ restricting to
$\pi_*\mathcal{O}_V$ over $U$. This proves (1).
\medskip\noindent
By construction $Y \times_X U = V$.
Since $\mathcal{B}$ is coherent, we see that $Y \to X$ is finite.
We have $\text{depth}(\mathcal{B}_x) \geq 2$ for $x \in X \setminus U$
by Divisors, Lemma \ref{divisors-lemma-reflexive-S2}.
Hence $\text{depth}(\mathcal{O}_{Y, y}) \geq 2$ for $y \in Y \setminus V$
by Algebra, Lemma \ref{algebra-lemma-depth-goes-down-finite}.
Conversely, suppose that $\pi' : Y' \to X$ is a finite morphism such that
$V = Y' \times_X U$ and $\text{depth}(\mathcal{O}_{Y', y'}) \geq 2$
for $y' \in Y' \setminus V$. Then $\pi'_*\mathcal{O}_{Y'}$
restricts to $\pi_*\mathcal{O}_V$ over $U$ and satisfies
$\text{depth}((\pi'_*\mathcal{O}_{Y'})_x) \geq 2$ for
$x \in X \setminus U$ by
Algebra, Lemma \ref{algebra-lemma-depth-goes-down-finite}.
Then $\pi'_*\mathcal{O}_{Y'}$ is canonically isomorphic
to $j_*\pi_*\mathcal{O}_V$ for example by
Divisors, Lemma \ref{divisors-lemma-depth-2-hartog}.
This proves (2).
\medskip\noindent
If $Y \to X$ is \'etale at $y$, then $Y \to X$ is flat at $y$.
Conversely, suppose that $Y \to X$ is flat at $y$.
If $y \in V$, then $Y \to X$ is \'etale at $y$.
If $y \not \in V$, then we check (1), (2), (3), and (4) of
Lemma \ref{lemma-ramification-quasi-finite-flat} hold
to see that $Y \to X$ is \'etale at $y$. Parts (1) and (2)
are clear and so is (3) since $\text{depth}(\mathcal{O}_{Y, y}) \geq 2$.
If $y' \leadsto y$ is a specialization and $\dim(\mathcal{O}_{Y, y'}) = 1$,
then $y' \in V$ since otherwise the depth of this local ring
would be $2$ a contradiction by
Algebra, Lemma \ref{algebra-lemma-bound-depth}.
Hence $Y \to X$ is \'etale at $y'$ and we conclude (4) of
Lemma \ref{lemma-ramification-quasi-finite-flat} holds too.
This finishes the proof of (3).
\medskip\noindent
Part (4) follows from (3) and the fact that $((Y \to X)_*\mathcal{O}_Y)_x$
is a flat $\mathcal{O}_{X, x}$-module if and only if $\mathcal{O}_{Y, y}$
is a flat $\mathcal{O}_{X, x}$-module for all $y \in Y$ mapping to $x$, see
Algebra, Lemma \ref{algebra-lemma-flat-localization}. Here we also
use that a finite flat module over a Noetherian ring is finite locally
free, see Algebra, Lemma \ref{algebra-lemma-finite-projective}
(and
Algebra, Lemma
\ref{algebra-lemma-Noetherian-finite-type-is-finite-presentation}).
\medskip\noindent
The final assertion of the lemma follows from
flat base change, see
Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}.
\end{proof}
@@ -5297,8 +5382,8 @@ \section{Finite \'etale covers of punctured spectra, IV}
\label{section-pi1-punctured-spec-IV}
\noindent
In this section we study when in Situation \ref{situation-local-lefschetz}.
the restriction functor
Let $X, X_0, U, U_0$ be as in Situation \ref{situation-local-lefschetz}.
In this section we ask when the restriction functor
$$
\textit{F\'Et}_U
\longrightarrow
@@ -5492,6 +5577,259 @@ \section{Purity in local case, II}
\section{Purity in local case, III}
\label{section-local-purity-III}
\noindent
In this section is a continuation of the discussion in
Sections \ref{section-local-purity} and \ref{section-local-purity-II}.
\begin{lemma}
\label{lemma-purity-power-series-over-depth2}
Let $(A, \mathfrak m)$ be a Noetherian local ring of depth $\geq 2$. Let
$B = A[[x_1, \ldots, x_d]]$ with $d \geq 1$. For any open
$V \subset Y = \Spec(B)$ which contains
\begin{enumerate}
\item any prime $\mathfrak q \subset B$ such that
$\mathfrak q \cap A \not = \mathfrak m$,
\item the prime $\mathfrak m B$
\end{enumerate}
the functor
$
\textit{F\'Et}_Y
\to
\textit{F\'Et}_V
$
is an equivalence. In particular purity holds for $B$.
\end{lemma}
\begin{proof}
A prime $\mathfrak q \subset B$ which is not contained in $V$
lies over $\mathfrak m$. In this case $A \to B_\mathfrak q$
is a flat local homomorphism and hence $\text{depth}(B_\mathfrak q) \geq 2$
(Algebra, Lemma \ref{algebra-lemma-apply-grothendieck}).
Thus the functor is fully faithful by
Lemma \ref{lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian}
combined with
Lemma \ref{lemma-depth-2-connected-punctured-spectrum}.
\medskip\noindent
Let $W \to V$ be a finite \'etale morphism. Let $B \to C$ be the unique finite
ring map such that $\Spec(C) \to Y$ is the finite morphism extending
$W \to V$ constructed in Lemma \ref{lemma-extend-S2}.
Set $I = (x_1, \ldots, x_d) \subset B$. For every $n \geq 0$ denote
$Y_n = \Spec(B/I^{n + 1})$ and consider the finite \'etale morphism
$$
W_n = Y_n \times_Y W \longrightarrow V_n = Y_n \cap V
$$
Set $X = \Spec(A)$ and denote $U \subset X$ the punctured spectrum of $A$.
Observe that $Y_0 \to X$ and $V_0 \to U$ are isomorphisms and
for $n \geq 1$ we have that
$$
V_n \cong U \times_X Y_n
$$
is a thickening of $V_0 = U$. By the topological invariance of
finite \'etale coverings we see that there are isomorphisms
$$
W_n \cong W_0 \times_X Y_n
$$
compatible for varying $n$ as both sides are schemes finite \'etale
over $V_n$ whose restriction to $V_0 = U$ is equal to $W_0$.
See Lemma \ref{lemma-thickening}. We conclude that
$$
\lim \Gamma(W_n, \mathcal{O}_{W_n}) =
\lim \Gamma(W_0, \mathcal{O}_{W_0}) \otimes_A B/I^n
$$
Observe that $C_0 = \Gamma(W_0, \mathcal{O}_{W_0})$ is a finite $A$-algebra
by Lemma \ref{lemma-extend-S2} applied to $W_0 \to U \subset X$ (exactly
as we did for $B \to C$ above). Thus we can consider the map
$$
C = \Gamma(W, \mathcal{O}_W)
\longrightarrow
\lim \Gamma(W_n, \mathcal{O}_{W_n}) = \lim C_0 \otimes_A B/I^n
= C_0 \otimes_A B
$$
The final equality on the right hand side holds by
Algebra, Proposition \ref{algebra-proposition-fp-tensor}.
This map determines a morphism
$$
\Psi : \Spec(C_0) \times_X Y \longrightarrow \Spec(C)
$$
of schemes over $Y$. Over $U \times_X Y$ these schemes are finite \'etale
and above points of $V_0 = U \times_X Y_0$
we obtain an isomorphism by construction.
Since the fibres of $U \times_X Y \to U$ are connected
(small detail omitted) we conclude that
$\Psi$ is an isomorphism over $U \times_X Y$.
Since the construction in Lemma \ref{lemma-extend-S2}
is compatible with the flat base change $Y \to X$,
we conclude that $\Psi$ must be an isomorphism
(by the uniqueness inherent in the lemma).
\medskip\noindent
However, we know that $\Spec(C) \to Y$ is \'etale at all points above
at least one point of $Y$ lying over $\mathfrak m \in X$.
Since $\Psi$ is an isomorphism, we conclude that $\Spec(C_0) \to X$
is \'etale at all points above $\mathfrak m$ (small detail omitted).
Of course this means that $A \to C_0$ is finite
\'etale and hence $B \to C$ is finite \'etale.
\end{proof}
\begin{lemma}
\label{lemma-purity-smooth-over-depth2}
Let $f : X \to S$ be a morphism of schemes. Let $U \subset X$
be an open subscheme. Assume
\begin{enumerate}
\item $f$ is smooth,
\item $S$ is Noetherian,
\item for $s \in S$ with $\text{depth}(\mathcal{O}_{S, s}) \leq 1$
we have $X_s = U_s$,
\item $U_s \subset X_s$ is dense for all $s \in S$.
\end{enumerate}
Then $\textit{F\'Et}_X \to \textit{F\'Et}_U$ is an equivalence.
\end{lemma}
\begin{proof}
The functor is fully faithful by
Lemma \ref{lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian}
combined with
Lemma \ref{lemma-depth-2-connected-punctured-spectrum}
(plus an application of
Algebra, Lemma \ref{algebra-lemma-apply-grothendieck}
to check the depth condition).
\medskip\noindent
Let $\pi : V \to U$ be a finite \'etale morphism. Let $Y \to X$
be the finite morphism constructed in Lemma \ref{lemma-extend-S2}.
We have to show that $Y \to X$ is finite \'etale.
To show that this is true for all points $x \in X$ mapping to a
given point $s \in S$ we may perform a base change by a flat
morphism $S' \to S$ of Noetherian schemes such that $s$ is
in the image. This follows from the compatibility of the
construction in Lemma \ref{lemma-extend-S2} with flat base change.
\medskip\noindent
After enlarging $U$ we may assume $U \subset X$ is
the maximal open over which $Y \to X$ is finite \'etale.
Let $Z \subset X$ be the complement of $U$.
To get a contradiction, assume $Z \not = \emptyset$.
Let $s \in S$ be a point in the image of $Z \to S$
such that no strict generalization of $s$ is in the image.
Then after base change to $\Spec(\mathcal{O}_{S, s})$
we see that $S = \Spec(A)$ with $(A, \mathfrak m, \kappa)$
a local Noetherian ring of depth $\geq 2$ and $Z$
contained in the closed fibre $X_s$
and nowhere dense in $X_s$. Choose a closed point $z \in Z$.
Then $\kappa(z)/\kappa$ is finite (by the Hilbert Nullstellensatz, see
Algebra, Theorem \ref{algebra-theorem-nullstellensatz}).
Choose a finite flat morphism $(S', s') \to (S, s)$ of local schemes
realizing the residue field extension $\kappa(z)/\kappa$, see
Algebra, Lemma \ref{algebra-lemma-finite-free-given-residue-field-extension}.
After doing a base change by $S' \to S$ we reduce to the case
where $\kappa(z) = \kappa$.
\medskip\noindent
By More on Morphisms, Lemma \ref{more-morphisms-lemma-slice-smooth}
there exists a locally closed subscheme $S' \subset X$ passing through $z$
such that $S' \to S$ is \'etale at $z$. After performing the base change
by $S' \to S$, we may assume there is a section $\sigma : S \to X$
such that $\sigma(s) = z$. Choose an affine neighbourhood
$\Spec(B) \subset X$ of $s$. Then $A \to B$ is a smooth ring
map which has a section $\sigma : B \to A$. Denote $I = \Ker(\sigma)$
and denote $B^\wedge$ the $I$-adic completion of $B$.
Then $B^\wedge \cong A[[x_1, \ldots, x_d]]$ for some $d \geq 0$, see
Algebra, Lemma \ref{algebra-lemma-section-smooth}.
Observe that $d > 0$ since otherwise we see that $X \to S$
is \'etale at $z$ which would imply that $z$ is a generic point of
$X_s$ and hence $z \in U$ by assumption (4).
Similarly, if $d > 0$, then $\mathfrak m B^\wedge$ maps into
$U$ via the morphism $\Spec(B^\wedge) \to X$.
It suffices prove $Y \to X$ is finite \'etale after base change
to $\Spec(B^\wedge)$. Since $B \to B^\wedge$ is flat
(Algebra, Lemma \ref{algebra-lemma-completion-flat})
this follows from Lemma \ref{lemma-purity-power-series-over-depth2}
and the uniqueness in the construction of $Y \to X$.
\end{proof}
\begin{proposition}
\label{proposition-purity-smooth-over-depth2}
Let $A \to B$ be a local homomorphism of local Noetherian rings.
Assume $A$ has depth $\geq 2$, $A \to B$ is formally smooth for the
$\mathfrak m_B$-adic topology, and $\dim(B) > \dim(A)$. For any open
$V \subset Y = \Spec(B)$ which contains
\begin{enumerate}
\item any prime $\mathfrak q \subset B$ such that
$\mathfrak q \cap A \not = \mathfrak m_A$,
\item the prime $\mathfrak m_A B$
\end{enumerate}
the functor $\textit{F\'Et}_Y \to \textit{F\'Et}_V$
is an equivalence. In particular purity holds for $B$.
\end{proposition}
\begin{proof}
A prime $\mathfrak q \subset B$ which is not contained in $V$
lies over $\mathfrak m_A$. In this case $A \to B_\mathfrak q$
is a flat local homomorphism and hence $\text{depth}(B_\mathfrak q) \geq 2$
(Algebra, Lemma \ref{algebra-lemma-apply-grothendieck}).
Thus the functor is fully faithful by
Lemma \ref{lemma-quasi-compact-dense-open-connected-at-infinity-Noetherian}
combined with
Lemma \ref{lemma-depth-2-connected-punctured-spectrum}.
\medskip\noindent
Denote $A^\wedge$ and $B^\wedge$ the completions of $A$ and $B$
with respect to their maximal ideals. Observe that the assumptions
of the proposition hold for $A^\wedge \to B^\wedge$, see
More on Algebra, Lemmas
\ref{more-algebra-lemma-completion-dimension},
\ref{more-algebra-lemma-completion-depth}, and
\ref{more-algebra-lemma-formally-smooth-completion}.
By the uniqueness and compatibility with flat base change
of the construction of Lemma \ref{lemma-extend-S2}
it suffices to prove the essential surjectivity for
$A^\wedge \to B^\wedge$ and the inverse image of $V$
(details omitted; compare with Lemma \ref{lemma-purity-and-completion}
for the case where $V$ is the punctured spectrum).
By More on Algebra, Proposition \ref{more-algebra-proposition-fs-regular}
this means we may assume $A \to B$ is regular.
\medskip\noindent
Let $W \to V$ be a finite \'etale morphism.
By Popescu's theorem
(Smoothing Ring Maps, Theorem \ref{smoothing-theorem-popescu})
we can write $B = \colim B_i$ as a filtered colimit
of smooth $A$-algebras. We can pick an $i$ and an
open $V_i \subset \Spec(B_i)$ whose inverse image is $V$
(Limits, Lemma \ref{limits-lemma-descend-opens}).
After increasing $i$ we may assume there is a finite
\'etale morphism $W_i \to V_i$ whose base change to $V$
is $W \to V$, see
Limits, Lemmas \ref{limits-lemma-descend-finite-presentation},
\ref{limits-lemma-descend-finite-finite-presentation}, and
\ref{limits-lemma-descend-etale}.
We may assume the complement of $V_i$ is contained
in the closed fibre of $\Spec(B_i) \to \Spec(A)$ as this
is true for $V$ (either choose $V_i$ this way or use
the lemma above to show this is true for $i$ large enough).
Let $\eta$ be the generic point of the closed fibre
of $\Spec(B) \to \Spec(A)$. Since $\eta \in V$, the image of
$\eta$ is in $V_i$. Hence after replacing $V_i$ by an
affine open neighbourhood of the image of the closed point
of $\Spec(B)$, we may assume that the closed fibre
of $\Spec(B_i) \to \Spec(A)$ is irreducible and that
its generic point is contained in $V_i$ (details omitted; use that
a scheme smooth over a field is a disjoint union of irreducible schemes).
At this point we may apply Lemma \ref{lemma-purity-smooth-over-depth2}
to see that $W_i \to V_i$ extends to a finite \'etale
morphism $\Spec(C_i) \to \Spec(B_i)$ and pulling
back to $\Spec(B)$ we conclude that $W$ is in
the essential image of the functor
$\textit{F\'Et}_Y \to \textit{F\'Et}_V$
as desired.
\end{proof}
\section{Lefschetz for the fundamental group}

0 comments on commit f43c5f2

Please sign in to comment.