# stacks/stacks-project

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.
 @@ -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}