From ea1587e98e32b3d17b0226e629c957d82f7fd2cf Mon Sep 17 00:00:00 2001 From: Aise Johan de Jong Date: Thu, 12 May 2016 10:18:31 -0400 Subject: [PATCH] More on normal crossings divisors --- etale.tex | 366 ++++++++++++++++++++++++++++++++++++++++++++++++++++ pione.tex | 13 -- resolve.tex | 86 +----------- tags/tags | 6 +- 4 files changed, 373 insertions(+), 98 deletions(-) diff --git a/etale.tex b/etale.tex index 78d8ba0fc..e58cd267a 100644 --- a/etale.tex +++ b/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} diff --git a/pione.tex b/pione.tex index 1c21f7218..e341366dc 100644 --- a/pione.tex +++ b/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 $(x - iy)(x + iy) = 0$. diff --git a/resolve.tex b/resolve.tex index cd11cc5b8..b638f91a6 100644 --- a/resolve.tex +++ b/resolve.tex @@ -4161,86 +4161,6 @@ \section{Embedded resolution} When our curve is contained on a regular surface we often want to turn it into a divisor with normal crossings. -\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 and for every nonempty finite subset $J \subset I$ -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-turn-into-effective-Cartier} Let $X$ be a regular scheme of dimension $2$. Let $Z \subset X$ @@ -4315,8 +4235,10 @@ \section{Embedded resolution} is regular, or (c) exactly two curves $Y_i$, $Y_j$ passing through $p$, the local rings $\mathcal{O}_{Y_i, p}$, $\mathcal{O}_{Y_j, p}$ are regular and $m_p(Y_i \cap Y_j) = 1$. -This exactly means that $\sum Y_i$ is a strict normal crossings -divisor on the regular surface $X$. +This means that $\sum Y_i$ is a strict normal crossings +divisor on the regular surface $X$, see +\'Etale Morphisms, Lemma +\ref{etale-lemma-strict-normal-crossings}. \end{proof} diff --git a/tags/tags b/tags/tags index 7dbee8378..f4cf8fe08 100644 --- a/tags/tags +++ b/tags/tags @@ -14477,8 +14477,8 @@ 0BI6,resolve-equation-multiplicity 0BI7,resolve-lemma-blowup-nonsingular-curves-meeting-at-point 0BI8,resolve-lemma-blowup-curves -0BI9,resolve-definition-strict-normal-crossings -0BIA,resolve-lemma-strict-normal-crossings +0BI9,etale-definition-strict-normal-crossings +0BIA,etale-lemma-strict-normal-crossings 0BIB,resolve-lemma-turn-into-effective-Cartier 0BIC,resolve-lemma-embedded-resolution 0BID,fields-lemma-subalgebra-algebraic-extension-field @@ -14798,7 +14798,7 @@ 0BSC,pione-lemma-exact-sequence-finite-nr-closed-pts 0BSD,pione-section-ramification 0BSE,pione-section-tame -0BSF,pione-definition-normal-crossings +0BSF,etale-definition-normal-crossings 0BSG,algebra-section-ind-etale 0BSH,algebra-lemma-base-change-colimit-etale 0BSI,algebra-lemma-composition-colimit-etale