More on normal crossings divisors

etale.tex

@@ -2360,6 +2360,372 @@ \section{Descending \'etale morphisms}
 
 
 
+\section{Normal crossings divisors}
+\label{section-normal-crossings}
+
+\noindent
+Here is the definition.
+
+\begin{definition}
+\label{definition-strict-normal-crossings}
+Let $X$ be a locally Noetherian scheme. A
+{\it strict normal crossings divisor}
+on $X$ is an effective Cartier divisor $D \subset X$ such that
+for every $p \in D$ the local ring $\mathcal{O}_{X, p}$ is regular
+and there exists a regular system of parameters
+$x_1, \ldots, x_d \in \mathfrak m_p$ and $1 \leq r \leq d$
+such that $D$ is cut out by $x_1 \ldots x_r$ in $\mathcal{O}_{X, p}$.
+\end{definition}
+
+\noindent
+We often encounter effective Cartier divisors $E$ on locally Noetherian
+schemes $X$ such that there exists a strict normal crossings divisor $D$
+with $E \subset D$ set theoretically.
+In this case we have
+$E = \sum a_i D_i$ with $a_i \geq 0$ where $D = \bigcup_{i \in I} D_i$
+is the decomposition of $D$ into its irreducible components.
+Observe that $D' = \bigcup_{a_i > 0} D_i$ is a strict normal crossings
+divisor with $E = D'$ set theoretically.
+When the above happens we will say that
+$E$ is {\it supported on a strict normal crossings divisor}.
+
+\begin{lemma}
+\label{lemma-strict-normal-crossings}
+Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be an
+effective Cartier divisor. Let $D_i \subset D$, $i \in I$ be its
+irreducible components viewed as reduced closed subschemes of $X$.
+The following are equivalent
+\begin{enumerate}
+\item $D$ is a strict normal crossings divisor, and
+\item $D$ is reduced, each $D_i$ is an effective Cartier divisor, and
+for $J \subset I$ finite the scheme theoretic
+intersection $D_J = \bigcap_{j \in J} D_j$ is a
+regular scheme each of whose irreducible components has
+codimension $|J|$ in $X$.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+Assume $D$ is a strict normal crossings divisor. Pick $p \in D$
+and choose a regular system of parameters $x_1, \ldots, x_d \in \mathfrak m_p$
+and $1 \leq r \leq d$ as in
+Definition \ref{definition-strict-normal-crossings}.
+Since $\mathcal{O}_{X, p}/(x_i)$ is a regular local ring
+(and in particular a domain) we see that the irreducible components
+$D_1, \ldots, D_r$ of $D$ passing through $p$ correspond $1$-to-$1$
+to the height one primes $(x_1), \ldots, (x_r)$ of $\mathcal{O}_{X, p}$.
+By Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}
+we find that the intersections $D_{i_1} \cap \ldots \cap D_{i_s}$
+have codimension $s$ in an open neighbourhood of $p$
+and that this intersection has a regular local ring at $p$.
+Since this holds for all $p \in D$ we conclude that (2) holds.
+
+\medskip\noindent
+Assume (2). Let $p \in D$. Since $\mathcal{O}_{X, p}$ is finite
+dimensional we see that $p$ can be contained in at most
+$\dim(\mathcal{O}_{X, p})$ of the components $D_i$.
+Say $p \in D_1, \ldots, D_r$ for some $r \geq 1$.
+Let $x_1, \ldots, x_r \in \mathfrak m_p$ be local equations
+for $D_1, \ldots, D_r$. Then $x_1$ is a nonzerodivisor in $\mathcal{O}_{X, p}$
+and $\mathcal{O}_{X, p}/(x_1) = \mathcal{O}_{D_1, p}$ is regular.
+Hence $\mathcal{O}_{X, p}$ is regular, see
+Algebra, Lemma \ref{algebra-lemma-regular-mod-x}.
+Since $D_1 \cap \ldots \cap D_r$ is a regular (hence normal) scheme
+it is a disjoint union of its irreducible components
+(Properties, Lemma \ref{properties-lemma-normal-Noetherian}).
+Let $Z \subset D_1 \cap \ldots \cap D_r$
+be the irreducible component containing $p$.
+Then $\mathcal{O}_{Z, p} = \mathcal{O}_{X, p}/(x_1, \ldots, x_r)$
+is regular of codimension $r$ (note that since we already know
+that $\mathcal{O}_{X, p}$ is regular and hence Cohen-Macaulay,
+there is no ambiguity about codimension as the ring is catenary, see
+Algebra, Lemmas \ref{algebra-lemma-regular-ring-CM} and
+\ref{algebra-lemma-CM-dim-formula}).
+Hence $\dim(\mathcal{O}_{Z, p}) = \dim(\mathcal{O}_{X, p}) - r$.
+Choose additional $x_{r + 1}, \ldots, x_n \in \mathfrak m_p$
+which map to a minimal system of generators of $\mathfrak m_{Z, p}$.
+Then $\mathfrak m_p = (x_1, \ldots, x_n)$ by Nakayama's lemma
+and we see that $D$ is a normal crossings divisor.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-smooth-pullback-strict-normal-crossings}
+\begin{slogan}
+Pullback of a strict normal crossings divisor by a smooth
+morphism is a strict normal crossings divisor.
+\end{slogan}
+Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a
+strict normal crossings divisor. If $f : Y \to X$ is a smooth
+morphism of schemes, then the pullback $f^*D$ is a
+strict normal crossings divisor on $Y$.
+\end{lemma}
+
+\begin{proof}
+As $f$ is flat the pullback is defined by
+Divisors, Lemma \ref{divisors-lemma-pullback-effective-Cartier-defined}
+hence the statement makes sense.
+Let $q \in f^*D$ map to $p \in D$. Choose a regular system
+of parameters $x_1, \ldots, x_d \in \mathfrak m_p$
+and $1 \leq r \leq d$ as in
+Definition \ref{definition-strict-normal-crossings}.
+Since $f$ is smooth the local ring homomorphism
+$\mathcal{O}_{X, p} \to \mathcal{O}_{Y, q}$ is flat
+and the fibre ring
+$$
+\mathcal{O}_{Y, q}/\mathfrak m_p \mathcal{O}_{Y, q} =
+\mathcal{O}_{Y_p, q}
+$$
+is a regular local ring (see for example
+Algebra, Lemma \ref{algebra-lemma-characterize-smooth-over-field}).
+Pick $y_1, \ldots, y_n \in \mathfrak m_q$ which map to a regular
+system of parameters in $\mathcal{O}_{Y_p, q}$.
+Then $x_1, \ldots, x_d, y_1, \ldots, y_n$ generate the
+maximal ideal $\mathfrak m_q$. Hence $\mathcal{O}_{Y, q}$
+is a regular local ring of dimension
+$d + n$ by Algebra, Lemma \ref{algebra-lemma-dimension-base-fibre-equals-total}
+and $x_1, \ldots, x_d, y_1, \ldots, y_n$
+is a regular system of parameters. Since $f^*D$ is cut
+out by $x_1 \ldots x_r$ in $\mathcal{O}_{Y, q}$ we conclude
+that the lemma is true.
+\end{proof}
+
+\noindent
+Here is the definition of a normal crossings divisor.
+
+\begin{definition}
+\label{definition-normal-crossings}
+Let $X$ be a locally Noetherian scheme. A {\it normal crossings divisor}
+on $X$ is an effective Cartier divisor $D \subset X$ such that for
+every $p \in D$ there exists an \'etale morphism $U \to X$ with
+$p$ in the image and $D \times_X U$ a
+strict normal crossings divisor on $U$.
+\end{definition}
+
+\noindent
+For example $D = V(x^2 + y^2)$ is a normal crossings divisor
+(but not a strict one) on
+$\Spec(\mathbf{R}[x, y])$ because after pulling back to
+the \'etale cover $\Spec(\mathbf{C}[x, y])$ we obtain $(x - iy)(x + iy) = 0$.
+
+\begin{lemma}
+\label{lemma-smooth-pullback-normal-crossings}
+\begin{slogan}
+Pullback of a normal crossings divisor by a smooth
+morphism is a normal crossings divisor.
+\end{slogan}
+Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a
+normal crossings divisor. If $f : Y \to X$ is a smooth
+morphism of schemes, then the pullback $f^*D$ is a
+normal crossings divisor on $Y$.
+\end{lemma}
+
+\begin{proof}
+As $f$ is flat the pullback is defined by
+Divisors, Lemma \ref{divisors-lemma-pullback-effective-Cartier-defined}
+hence the statement makes sense.
+Let $q \in f^*D$ map to $p \in D$.
+Choose an \'etale morphism $U \to X$ whose image contains $p$
+such that $D \times_X U \subset U$ is a strict normal crossings
+divisor as in Definition \ref{definition-normal-crossings}.
+Set $V = Y \times_X U$. Then $V \to Y$ is \'etale as a base
+change of $U \to X$
+(Morphisms, Lemma \ref{morphisms-lemma-base-change-etale})
+and the pullback $D \times_X V$ is a strict normal crossings
+divisor on $V$ by Lemma \ref{lemma-smooth-pullback-strict-normal-crossings}.
+Thus we have checked the condition of
+Definition \ref{definition-normal-crossings}
+for $q \in f^*D$ and we conclude.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-characterize-normal-crossings-normalization}
+Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a closed
+subscheme. The following are equivalent
+\begin{enumerate}
+\item $D$ is a normal crossings divisor in $X$,
+\item $D$ is reduced, the normalization $\nu : D^\nu \to D$ is unramified,
+and for any $n \geq 1$ the scheme
+$$
+Z_n = D^\nu \times_D \ldots \times_D D^\nu
+\setminus \{(p_1, \ldots, p_n) \mid p_i = p_j\text{ for some }i\not = j\}
+$$
+is regular, the morphism $Z_n \to X$ is a local complete intersection
+morphism whose conormal sheaf is locally free of rank $n$.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+First we explain how to think about condition (2).
+The diagonal of an unramified morphism is open
+(Morphisms, Lemma \ref{morphisms-lemma-diagonal-unramified-morphism}).
+On the other hand $D^\nu \to D$ is separated, hence the
+diagonal $D^\nu \to D^\nu \times_D D^\nu$ is closed.
+Thus $Z_n$ is an open and closed subscheme of
+$D^\nu \times_D \ldots \times_D D^\nu$. On the other hand,
+$Z_n \to X$ is unramified as it is the compostion
+$$
+Z_n \to D^\nu \times_D \ldots \times_D D^\nu \to \ldots \to
+D^\nu \times_D D^\nu \to D^\nu \to D \to X
+$$
+and each of the arrows is unramified.
+Since an unramified morphism is formally unramified
+(More on Morphisms, Lemma
+\ref{more-morphisms-lemma-unramified-formally-unramified})
+we have a conormal sheaf
+$\mathcal{C}_n = \mathcal{C}_{Z_n/X}$ of $Z_n \to X$, see
+More on Morphisms, Definition
+\ref{more-morphisms-definition-universal-thickening}.
+
+\medskip\noindent
+Formation of normalization commutes with \'etale localization by
+More on Morphisms, Lemma \ref{more-morphisms-lemma-normalization-and-smooth}.
+Checking that local rings are regular, or that
+a morphism is unramified, or that a morphism is a
+local complete intersection or that a morphism is
+unramified and has a conormal sheaf which is
+locally free of a given rank, may be done \'etale locally (see
+More on Algebra, Lemma \ref{more-algebra-lemma-regular-etale-extension},
+Descent, Lemma \ref{descent-lemma-descending-property-unramified},
+More on Morphisms, Lemma \ref{more-morphisms-lemma-descending-property-lci}
+and
+Descent, Lemma \ref{descent-lemma-finite-locally-free-descends}).
+
+\medskip\noindent
+By the remark of the preceding paragraph and the definition
+of normal crossings divisor it suffices to prove that a
+strict normal crossings divisor $D = \bigcup_{i \in I} D_i$
+satisfies (2). In this case $D^\nu = \coprod D_i$
+and $D^\nu \to D$ is unramified (being unramified
+is local on the source and $D_i \to D$ is a closed
+immersion which is unramified). Simiarly, $Z_1 = D^\nu \to X$
+is a local complete intersection morphism because we may
+check this locally on the source and each morphism $D_i \to X$
+is a regular immersion as it is the inclusion of a Cartier divisor
+(see Lemma \ref{lemma-strict-normal-crossings} and
+More on Morphisms, Lemma \ref{more-morphisms-lemma-regular-immersion-lci}).
+Since an effective Cartier divisor has an invertible
+conormal sheaf, we conclude that the requirement on the
+conormal sheaf is satisfied.
+Similarly, the scheme $Z_n$ for $n \geq 2$ is the disjoint union
+of the schemes $D_J = \bigcap_{j \in J} D_j$ where $J \subset I$
+runs over the subsets of order $n$. Since $D_J \to X$ is
+a regular immersion of codimension $n$
+(by the definition of strict normal crossings and the
+fact that we may check this on stalks by
+Divisors, Lemma \ref{divisors-lemma-Noetherian-scheme-regular-ideal})
+it follows in the same manner that $Z_n \to X$ has the required
+properties. Some details omitted.
+
+\medskip\noindent
+Assume (2). Let $p \in D$. Since $D^\nu \to D$ is unramified, it is
+finite (by Morphisms, Lemma \ref{morphisms-lemma-finite-integral}).
+Hence $D^\nu \to X$ is finite unramified.
+By Lemma \ref{lemma-finite-unramified-etale-local}
+and \'etale localization (permissible by the discussion
+in the second paragraph and the definition of normal
+crossings divisors) we reduce to the case where
+$D^\nu = \coprod_{i \in I} D_i$
+with $I$ finite and $D_i \to U$ a closed immersion.
+After shrinking $X$ if necessary, we may assume
+$p \in D_i$ for all $i \in I$. The condition that $Z_1 =  D^\nu \to X$ is an
+unramified local complete intersection morphism
+with conormal sheaf locally free of rank $1$
+implies that $D_i \subset X$ is an effective Cartier divisor, see
+More on Morphisms, Lemma \ref{more-morphisms-lemma-lci} and
+Divisors, Lemma \ref{divisors-lemma-regular-immersion-noetherian}.
+To finish the proof we may assume $X = \Spec(A)$ is affine
+and $D_i = V(f_i)$ with $f_i \in A$ a nonzerodivisor.
+If $I = \{1, \ldots, r\}$, then $p \in Z_r = V(f_1, \ldots, f_r)$.
+The same reference as above implies that
+$(f_1, \ldots, f_r)$ is a Koszul regular ideal in $A$.
+Since the conormal sheaf has rank $r$, we see that
+$f_1, \ldots, f_r$ is a minimal set of generators of
+the ideal defining $Z_r$ in $\mathcal{O}_{X, p}$.
+This implies that $f_1, \ldots, f_r$ is a regular sequence
+in $\mathcal{O}_{X, p}$ such that $\mathcal{O}_{X, p}/(f_1, \ldots, f_r)$
+is regular. Thus we conclude by
+Algebra, Lemma \ref{algebra-lemma-regular-mod-x}
+that $f_1, \ldots, f_r$ can be extended to a regular system of parameters
+in $\mathcal{O}_{X, p}$ and this finishes the proof.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-characterize-normal-crossings}
+Let $X$ be a locally Noetherian scheme. Let $D \subset X$ be a closed
+subscheme. If $X$ is J-2 or Nagata, then following are equivalent
+\begin{enumerate}
+\item $D$ is a normal crossings divisor in $X$,
+\item for every $p \in D$ the pullback of $D$ to the spectrum of the
+strict henselization $\mathcal{O}_{X, p}^{sh}$
+is a strict normal crossings divisor.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+The implication (1) $\Rightarrow$ (2) is straightforward and
+does not need the assumption that $X$ is J-2 or Nagata.
+Namely, let $p \in D$ and choose an \'etale neighbourhood
+$(U, u) \to (X, p)$ such that the pullback of $D$ is
+a strict normal crossings divisor on $U$.
+Then $\mathcal{O}_{X, p}^{sh} = \mathcal{O}_{U, u}^{sh}$
+and we see that the trace of $D$ on $\Spec(\mathcal{O}_{U, u}^{sh})$
+is cut out by part of a regular system of parameters
+as this is already the case in $\mathcal{O}_{U, u}$.
+
+\medskip\noindent
+To prove the implication in the other direction
+we will use the criterion of
+Lemma \ref{lemma-characterize-normal-crossings-normalization}.
+Observe that formation of the normalization $D^\nu \to D$
+commutes with strict henselization, see
+More on Morphisms, Lemma
+\ref{more-morphisms-lemma-normalization-and-henselization}.
+If we can show that $D^\nu \to D$ is finite,
+then we see that $D^\nu \to D$ and the schemes
+$Z_n$ satisfy all desired properties because these
+can all be checked on the level of local rings
+(but the finiteness of the morphism $D^\nu \to D$
+is not something we can check on local rings).
+We omit the detailed verfications.
+
+\medskip\noindent
+If $X$ is Nagata, then $D^\nu \to D$ is finite by
+Morphisms, Lemma \ref{morphisms-lemma-nagata-normalization}.
+
+\medskip\noindent
+Assume $X$ is J-2. Choose a point $p \in D$. We will show
+that $D^\nu \to D$ is finite over a neighbourhood of $p$.
+By assumption there exists a regular system of
+parameters $f_1, \ldots, f_d$ of $\mathcal{O}_{X, p}^{sh}$
+and $1 \leq r \leq d$ such that the trace of $D$ on
+$\Spec(\mathcal{O}_{X, p}^{sh})$ is cut out by $f_1 \ldots f_r$.
+Then
+$$
+D^\nu \times_X \Spec(\mathcal{O}_{X, p}^{sh}) = 
+\coprod\nolimits_{i = 1, \ldots, r} V(f_i)
+$$
+Choose an affine \'etale neighbourhood
+$(U, u) \to (X, p)$ such that $f_i$ comes from
+$f_i \in \mathcal{O}_U(U)$. Set $D_i = V(f_i) \subset U$.
+The strict henselization of $\mathcal{O}_{D_i, u}$
+is $\mathcal{O}_{X, p}^{sh}/(f_i)$ which is regular.
+Hence $\mathcal{O}_{D_i, u}$ is regular (for example by
+More on Algebra, Lemma \ref{more-algebra-lemma-henselization-regular}).
+Because $X$ is J-2 the regular locus is open in $D_i$.
+Thus after replacing $U$ by a Zariski open we may assume
+that $D_i$ is regular for each $i$. It follows that
+$$
+\coprod\nolimits_{i = 1, \ldots, r} D_i = D^\nu \times_X U
+\longrightarrow D \times_X U
+$$
+is the normalization morphism and it is clearly finite.
+In other words, we have found
+an \'etale neighbourhood $(U, u)$ of $(X, p)$ such that
+the base change of $D^\nu \to D$ to this neighbourhood is finite.
+This implies $D^\nu \to D$ is finite by descent
+(Descent, Lemma \ref{descent-lemma-descending-property-finite})
+and the proof is complete.
+\end{proof}
 
 
 

pione.tex

@@ -5208,19 +5208,6 @@ \section{Tame ramification}
 as given in \cite{Grothendieck-Murre} but only in the case that $D$ is
 a divisor with normal crossings.
 
-\begin{definition}
-\label{definition-normal-crossings}
-Let $X$ be a locally Noetherian scheme. A {\it normal crossings divisor}
-on $X$ is an effective Cartier divisor $D \subset X$ such that there
-exists an \'etale covering $\{U_i \to X\}_{i \in I}$ with
-$D \times_X U_i \subset U_i$ is a strict normal crossings divisor
-for each $i$.
-\end{definition}
-
-\noindent
-For example $D = V(x^2 + y^2)$ is a normal crossings divisor on
-$\Spec(\mathbf{R}[x, y])$ because after pulling back to
-the \'etale cover $\Spec(\mathbf{C}[x, y])$ we obtain $(x - iy)(x + iy) = 0$.