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Abhyankar in global case of single divisor

The proof does not seem overly long... Perhaps the application of
the first Abhyankar lemma (from more-algebra) could be discussed in more
detail. Not sure. It might make things worse.
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aisejohan committed Sep 20, 2018
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321 pione.tex
@@ -5160,6 +5160,37 @@ \section{Purity of branch locus}
Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology}.
\end{proof}
\begin{lemma}
\label{lemma-extend-pure}
Let $j : U \to X$ be an open immersion of Noetherian schemes
such that purity holds for $\mathcal{O}_{X, x}$ for all $x \not \in U$.
Then
$$
\textit{F\'Et}_X \longrightarrow \textit{F\'Et}_U
$$
is essentially surjective.
\end{lemma}
\begin{proof}
Let $V \to U$ be a finite \'etale morphism. By Noetherian
induction it suffices to extend $V \to U$ to a finite \'etale
morphism to a strictly larger open subset of $X$.
Let $x \in X \setminus U$ be the generic point of
an irreducible component of $X \setminus U$.
Then the inverse image $U_x$ of $U$ in $\Spec(\mathcal{O}_{X, x})$
is the punctured spectrum of $\mathcal{O}_{X, x}$.
By assumption $V_x = V \times_U U_x$ is the restriction
of a finite \'etale morphism $Y_x \to \Spec(\mathcal{O}_{X, x})$
to $U_x$.
By Limits, Lemma \ref{limits-lemma-glueing-near-point}
we find an open subscheme $U \subset U' \subset X$
containing $x$ and a morphism $V' \to U'$ of finite presentation
whose restriction to $U$ recovers $V \to U$ and
whose restriction to $\Spec(\mathcal{O}_{X, x})$ recovering $Y_x$.
Finally, the morphism $V' \to U'$ is finite \'etale
after possible shrinking $U'$ to a smaller open by
Limits, Lemma \ref{limits-lemma-glueing-near-point-properties}.
\end{proof}
@@ -6576,6 +6607,12 @@ \section{Specialization maps in the smooth proper case}
\section{Tame ramification}
\label{section-tame}
@@ -6587,11 +6624,285 @@ \section{Tame ramification}
\medskip\noindent
In this section we discuss a different more elementary question which
precedes the notion of tameness at infinity. Namely, given a scheme
$X$ and a dense open $U \subset X$ when is a finite morphism $f : Y \to X$
tamely ramified relative to $D = X \setminus U$? We will use the definition
as given in \cite{Grothendieck-Murre} but only in the case that $D$ is
a divisor with normal crossings.
precedes the notion of tameness at infinity.
Please compare with the (slightly different)
discussion in \cite{Grothendieck-Murre}.
Assume we are given
\begin{enumerate}
\item a locally Noetherian scheme $X$,
\item a dense open $U \subset X$,
\item a finite \'etale morphism $f : Y \to U$
\end{enumerate}
such that for every for every prime divisor $Z \subset X$
with $Z \cap U = \emptyset$ the local ring $\mathcal{O}_{X, \xi}$
of $X$ at the generic point $\xi$ of $Z$ is a discrete valuation ring.
Setting $K_\xi$ equal to the fraction field of $\mathcal{O}_{X, \xi}$
we obtain a cartesian square
$$
\xymatrix{
\Spec(K_\xi) \ar[r] \ar[d] & U \ar[d] \\
\Spec(\mathcal{O}_{X, \xi}) \ar[r] & X
}
$$
of schemes. In particular, we see that $Y \times_U \Spec(K_\xi)$
is the spectrum of a finite separable algebra $L_\xi/K$.
Then we say
{\it $Y$ is unramified over $X$ in codimension $1$},
resp.\ {\it $Y$ is tamely ramified over $X$ in codimension $1$}
if $L_\xi/K_\xi$ is unramified, resp.\ tamely ramified
with respect to $\mathcal{O}_{X, \xi}$ for every $(Z, \xi)$
as above, see More on Algebra, Definition
\ref{more-algebra-definition-types-of-extensions}.
More precisely, we decompose $L_\xi$ into a product of finite
separable field extensions of $K_\xi$ and we require each of these
to be unramified, resp.\ tamely ramified with respect to
$\mathcal{O}_{X, \xi}$.
\begin{lemma}
\label{lemma-pullback-tame-codim1}
Let $X' \to X$ be a morphism of locally Noetherian schemes.
Let $U \subset X$ be a dense open. Assume
\begin{enumerate}
\item $U' = f^{-1}(U)$ is dense open in $X'$,
\item for every prime divisor $Z \subset X$ with $Z \cap U = \emptyset$
the local ring $\mathcal{O}_{X, \xi}$ of $X$ at the generic point $\xi$
of $Z$ is a discrete valuation ring,
\item for every prime divisor $Z' \subset X'$
with $Z' \cap U' = \emptyset$ the local ring $\mathcal{O}_{X', \xi'}$
of $X'$ at the generic point $\xi'$ of $Z'$ is a discrete valuation ring,
\item if $\xi' \in X'$ is as in (3), then $\xi = f(\xi')$ is as in (2).
\end{enumerate}
Then if $f : Y \to U$ is finite \'etale and
$Y$ is unramified, resp.\ tamely ramified over $X$
in codimension $1$, then $Y' = Y \times_X X' \to U'$ is finite \'etale
and $Y'$ is unramified, resp.\ tamely ramified over $X'$ in codimension $1$.
\end{lemma}
\begin{proof}
The only interesting fact in this lemma is the commutative algebra
result given in More on Algebra, Lemma \ref{more-algebra-lemma-tame-goes-up}.
\end{proof}
\noindent
Using the terminology introduced above, we can reformulate our
purity results obtained earlier in the following pleasing manner.
\begin{lemma}
\label{lemma-purity-one-divisor}
Let $X$ be a locally Noetherian scheme. Let $D \subset X$
be an effective Cartier divisor such that $D$ is a regular scheme.
Let $Y \to X \setminus D$ be a finite \'etale morphism.
If $Y$ is unramified over $X$ in codimension $1$, then
there exists a finite \'etale morphism $Y' \to X$
whose restriction to $X \setminus D$ is $Y$.
\end{lemma}
\begin{proof}
Before we start we note that $\mathcal{O}_{X, x}$ is a regular
local ring for all $x \in D$. This follows from
Algebra, Lemma \ref{algebra-lemma-regular-mod-x}
and our assumption that $\mathcal{O}_{D, x}$ is regular.
Let $\xi \in D$ be a generic point of an irreducible component of $D$.
By the above $\mathcal{O}_{X, \xi}$ is a discrete valuation ring.
Hence the statement of the lemma makes sense.
As in the discussion above, write
$Y \times_U \Spec(K_\xi) = \Spec(L_\xi)$.
Denote $B_\xi$ the integral closure of $\mathcal{O}_{X, \xi}$ in
$L_\xi$. Our assumption that $Y$ is unramified over $X$ in codimension $1$
signifies that $\mathcal{O}_{X, \xi} \to B_\xi$ is finite \'etale.
Thus we get $Y_\xi \to \Spec(\mathcal{O}_{X, \xi})$ finite
\'etale and an isomorphism
$$
Y \times_U \Spec(K_\xi) \cong
Y_\xi \times_{\Spec(\mathcal{O}_{X, \xi})} \Spec(K_\xi)
$$
over $\Spec(K_\xi)$.
By Limits, Lemma \ref{limits-lemma-glueing-near-point}
we find an open subscheme $X \setminus D \subset U' \subset X$
containing $\xi$ and a morphism $Y' \to U'$ of finite presentation
whose restriction to $X \setminus D$ recovers $Y$ and
whose restriction to $\Spec(\mathcal{O}_{X, \xi})$ recovers $Y_\xi$.
Finally, the morphism $Y' \to U'$ is finite \'etale
after possible shrinking $U'$ to a smaller open by
Limits, Lemma \ref{limits-lemma-glueing-near-point-properties}.
Repeating the argument with the other generic points of
$D$ we may assume that we have a finite \'etale morphism $Y' \to U'$
extending $Y \to X\setminus D$ to an open subscheme
containing $U' \subset X$ containing $X \setminus D$
and all generic points of $D$.
We finish by applying Lemma \ref{lemma-extend-pure}
to $Y' \to U'$. Namely, all local rings $\mathcal{O}_{X, x}$
for $x \in D$ are regular (see above) and if $x \not \in U'$
we have $\dim(\mathcal{O}_{X, x}) \geq 2$. Hence we have
purity for $\mathcal{O}_{X, x}$ by
Lemma \ref{lemma-local-purity}.
\end{proof}
\begin{example}[Standard tamely ramified morphism]
\label{example-tamely-ramified}
Let $A$ be a Noetherian ring. Let $f \in A$ be a nonzerodivisor
such that $A/fA$ is reduced. This implies that $A_\mathfrak p$
is a discrete valuation ring with uniformizer $f$ for any
minimal prime $\mathfrak p$ over $f$. Let $e \geq 1$ be an integer which
is invertible in $A$. Set
$$
C = A[x]/(x^e - f)
$$
Then $\Spec(C) \to \Spec(A)$ is a finite locally free morphism
which is \'etale over the spectrum of $A_f$. The finite \'etale morphism
$$
\Spec(C_f) \longrightarrow \Spec(A_f)
$$
is tamely ramified over $\Spec(A)$ in codimension $1$. The
tameness follows immediately from the characterization of tamely ramified
extensions in
More on Algebra, Lemma \ref{more-algebra-lemma-characterize-tame}.
\end{example}
\noindent
Here is a version of Abhyankar's lemma for regular divisors.
\begin{lemma}[Abhyankar's lemma for regular divisor]
\label{lemma-abhyankar-one-divisor}
Let $X$ be a locally Noetherian scheme. Let $D \subset X$
be an effective Cartier divisor such that $D$ is a regular scheme.
Let $Y \to X \setminus D$ be a finite \'etale morphism.
If $Y$ is tamely ramified over $X$ in codimension $1$, then
\'etale locally on $X$ the morphism $Y \to X$ is as given
as a finite disjoint union of standard tamely ramified
morphisms as described in Example \ref{example-tamely-ramified}.
\end{lemma}
\begin{proof}
Before we start we note that $\mathcal{O}_{X, x}$ is a regular
local ring for all $x \in D$. This follows from
Algebra, Lemma \ref{algebra-lemma-regular-mod-x}
and our assumption that $\mathcal{O}_{D, x}$ is regular.
Below we will also use that regular rings are normal, see
Algebra, Lemma \ref{algebra-lemma-regular-normal}.
\medskip\noindent
To prove the lemma we may work locally on $X$.
Thus we may assume $X = \Spec(A)$ and $D \subset X$
is given by a nonzerodivisor $f \in A$.
Then $Y = \Spec(B)$ as a finite \'etale scheme over $A_f$.
Let $\mathfrak p_1, \ldots, \mathfrak p_r$ be the minimal
primes of $A$ over $f$. Then $A_i = A_{\mathfrak p_i}$
is a discrete valuation ring; denote its fraction field $K_i$.
By assumption
$$
K_i \otimes_{A_f} B = \prod L_{ij}
$$
is a finite product of fields each tamely ramified with respect to $A_i$.
Choose $e \geq 1$ sufficiently divisible (namely, divisible by
all ramification indices for $L_{ij}$ over $A_i$ as in
More on Algebra, Remark \ref{more-algebra-remark-finite-separable-extension}).
Warning: at this point we do not know that $e$ is invertible on $A$.
\medskip\noindent
Consider the finite free $A$-algebra
$$
A' = A[x]/(x^e - f)
$$
Observe that $f' = x$ is a nonzerodivisor in $A'$ and that
$A'/f'A' \cong A/fA$ is a regular ring. Set
$B' = B \otimes_A A' = B \otimes_{A_f} A'_{f'}$.
By Abhyankar's lemma
(More on Algebra, Lemma \ref{more-algebra-lemma-abhyankar})
we see that $\Spec(B')$ is unramified over $\Spec(A')$
in codimension $1$. Namely, by Lemma \ref{lemma-pullback-tame-codim1}
we see that $\Spec(B')$ is still at least tamely ramified
over $\Spec(A')$ in codimension $1$. But Abhyankar's lemma
tells us that the ramification indices have all become equal to $1$.
By Lemma \ref{lemma-purity-one-divisor} we conclude that
$\Spec(B') \to \Spec(A'_{f'})$ extends to a finite \'etale morphism
$\Spec(C) \to \Spec(A')$.
\medskip\noindent
For a point $x \in D$ corresponding to $\mathfrak p \in V(f)$
denote $A^{sh}$ a strict henselization of $A_\mathfrak p = \mathcal{O}_{X, x}$.
Observe that $A^{sh}$ and $A^{sh}/fA^{sh} = (A/fA)^{sh}$
(Algebra, Lemma \ref{algebra-lemma-quotient-strict-henselization})
are regular local rings, see
More on Algebra, Lemma \ref{more-algebra-lemma-henselization-regular}.
Observe that $A'$ has a unique prime $\mathfrak p'$ lying over
$\mathfrak p$ with identical residue field. Thus
$$
(A')^{sh} = A^{sh} \otimes_A A' = A^{sh}[x]/(x^e - f)
$$
is a strictly henselian local ring finite over $A^{sh}$
(Algebra, Lemma \ref{algebra-lemma-quasi-finite-strict-henselization}).
Since $f'$ is a nonzerodivisor in $(A')^{sh}$ and since
$(A')^{sh}/f'(A')^{sh} = A^{sh}/fA^{sh}$ is regular, we conclude
that $(A')^{sh}$ is a regular local ring (see above).
Observe that the induced extension
$$
Q(A^{sh}) \subset Q((A')^{sh}) = Q(A^{sh})[x]/(x^e - f)
$$
of fraction fields has degree $e$ (and not less).
Since $A' \to C$ is finite \'etale we see that
$A^{sh} \otimes_A C$ is a finite product of copies of $(A')^{sh}$
(Algebra, Lemma \ref{algebra-lemma-mop-up-strictly-henselian}).
We have the inclusions
$$
A^{sh}_f \subset
A^{sh} \otimes_A B \subset
A^{sh} \otimes_A B' = A^{sh} \otimes_A C_{f'}
$$
and each of these rings is Noetherian and normal; this follows from
Algebra, Lemma \ref{algebra-lemma-normal-goes-up} for the ring
in the middle. Taking total quotient rings, using the product
decomposition of $A^{sh} \otimes_A C$ and using
Fields, Lemma \ref{fields-lemma-subfields-kummer} we conclude that
there is an isomorphism
$$
Q(A^{sh}) \otimes_A B \cong \prod\nolimits_{i \in I} F_i,\quad
F_i \cong Q(A^{sh})[x]/(x^{e_i} - f)
$$
of $Q(A^{sh})$-algebras for some finite set $I$ and integers $e_i | e$.
Since $A^{sh} \otimes_A B$ is a normal ring, it must be the
integral closure of $A^{sh}$ in its total quotient ring.
We conclude that we have an isomorphism
$$
A^{sh} \otimes_A B \cong \prod A^{sh}_f[x]/(x^{e_i} - f)
$$
over $A^{sh}_f$ because the algebras $A^{sh}[x]/(x^{e_i} - f)$
are regular and hence normal. The discriminant of
$A^{sh}[x]/(x^{e_i} - f)$ over $A^{sh}$ is $e_i^{e_i}f^{e_i - 1}$
(up to sign; calculation omitted). Since $A_f \to B$ is finite
\'etale we see that $e_i$ must be invertible in $A^{sh}_f$.
On the other hand, since $A_f \to B$ is tamely ramified
over $\Spec(A)$ in codimension $1$, by Lemma \ref{lemma-pullback-tame-codim1}
the ring map $A^{sh}_f \to A^{sh} \otimes_A B$
is tamely ramified over $\Spec(A^{sh})$ in codimension $1$.
This implies $e_i$ is nonzero in $A^{sh}/fA^{sh}$
(as it must map to an invertible element of the fraction field of
this domain by definition of tamely ramified extensions).
We conclude that $V(e_i) \subset \Spec(A^{sh})$
has codimension $\geq 2$ which is absurd unless it is empty.
In other words, $e_i$ is an invertible element of $A^{sh}$.
We conclude that the pullback of $Y$ to $\Spec(A^{sh})$
is indeed a finite disjoint union of standard tamely ramified morphisms.
\medskip\noindent
To finish the proof, we write $A^{sh} = \colim A_\lambda$
as a filtered colimit of \'etale $A$-algebras $A_\lambda$.
The isomorphism
$$
A^{sh} \otimes_A B \cong
\prod\nolimits_{i \in I} A^{sh}_f[x]/(x^{e_i} - f)
$$
descends to an isomorphism
$$
A_\lambda \otimes_A B \cong \prod\nolimits_{i \in I}
(A_\lambda)_f[x]/(x^{e_i} - f)
$$
for suitably large $\lambda$. After increasing $\lambda$ a bit
more we may assume $e_i$ is invertible in $A_\lambda$. Then
$\Spec(A_\lambda) \to \Spec(A)$ is the desired \'etale neighbourhood
of $x$ and the proof is complete.
\end{proof}

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