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 \input{preamble} % OK, start here. % \begin{document} \title{\'Etale Morphisms of Schemes} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this Chapter, we discuss \'etale morphisms of schemes. We illustrate some of the more important concepts by working with the Noetherian case. Our principal goal is to collect for the reader enough commutative algebra results to start reading a treatise on \'etale cohomology. An auxiliary goal is to provide enough evidence to ensure that the reader stops calling the phrase the \'etale topology of schemes'' an exercise in general nonsense, if (s)he does indulge in such blasphemy. \medskip\noindent We will refer to the other chapters of the Stacks project for standard results in algebraic geometry (on schemes and commutative algebra). We will provide detailed proofs of the new results that we state here. \section{Conventions} \label{section-conventions} \noindent In this chapter, frequently schemes will be assumed locally Noetherian and frequently rings will be assumed Noetherian. But in all the statements we will reiterate this when necessary, and make sure we list all the hypotheses! On the other hand, here are some general facts that we will use often and are useful to keep in mind: \begin{enumerate} \item A ring homomorphism $A \to B$ of finite type with $A$ Noetherian is of finite presentation. See Algebra, Lemma \ref{algebra-lemma-Noetherian-finite-type-is-finite-presentation}. \item A morphism (locally) of finite type between locally Noetherian schemes is automatically (locally) of finite presentation. See Morphisms, Lemma \ref{morphisms-lemma-noetherian-finite-type-finite-presentation}. \item Add more like this here. \end{enumerate} \section{Unramified morphisms} \label{section-unramified-definition} \noindent We first define unramified homomorphisms of local rings'' for Noetherian local rings. We cannot use the term unramified'' as there already is a notion of an unramified ring map (Algebra, Section \ref{algebra-section-unramified}) and it is different. After discussing the notion a bit we globalize it to describe unramified morphisms of locally Noetherian schemes. \begin{definition} \label{definition-unramified-rings} Let $A$, $B$ be Noetherian local rings. A local homomorphism $A \to B$ is said to be {\it unramified homomorphism of local rings} if \begin{enumerate} \item $\mathfrak m_AB = \mathfrak m_B$, \item $\kappa(\mathfrak m_B)$ is a finite separable extension of $\kappa(\mathfrak m_A)$, and \item $B$ is essentially of finite type over $A$ (this means that $B$ is the localization of a finite type $A$-algebra at a prime). \end{enumerate} \end{definition} \noindent This is the local version of the definition in Algebra, Section \ref{algebra-section-unramified}. In that section a ring map $R \to S$ is defined to be unramified if and only if it is of finite type, and $\Omega_{S/R} = 0$. We say $R \to S$ is unramified at a prime $\mathfrak q \subset S$ if there exists a $g \in S$, $g \not \in \mathfrak q$ such that $R \to S_g$ is an unramified ring map. It is shown in Algebra, Lemmas \ref{algebra-lemma-unramified-at-prime} and \ref{algebra-lemma-characterize-unramified} that given a ring map $R \to S$ of finite type, and a prime $\mathfrak q$ of $S$ lying over $\mathfrak p \subset R$, then we have $$R \to S\text{ is unramified at }\mathfrak q \Leftrightarrow \mathfrak pS_{\mathfrak q} = \mathfrak q S_{\mathfrak q} \text{ and } \kappa(\mathfrak p) \subset \kappa(\mathfrak q)\text{ finite separable}$$ Thus we see that for a local homomorphism of local rings the properties of our definition above are closely related to the question of being unramified. In fact, we have proved the following lemma. \begin{lemma} \label{lemma-characterize-unramified-Noetherian} \begin{slogan} Unramifiedness is a stalk local condition. \end{slogan} Let $A \to B$ be of finite type with $A$ a Noetherian ring. Let $\mathfrak q$ be a prime of $B$ lying over $\mathfrak p \subset A$. Then $A \to B$ is unramified at $\mathfrak q$ if and only if $A_{\mathfrak p} \to B_{\mathfrak q}$ is an unramified homomorphism of local rings. \end{lemma} \begin{proof} See discussion above. \end{proof} \noindent We will characterize the property of being unramified in terms of completions. For a Noetherian local ring $A$ we denote $A^\wedge$ the completion of $A$ with respect to the maximal ideal. It is also a Noetherian local ring, see Algebra, Lemma \ref{algebra-lemma-completion-Noetherian-Noetherian}. \begin{lemma} \label{lemma-unramified-completions} Let $A$, $B$ be Noetherian local rings. Let $A \to B$ be a local homomorphism. \begin{enumerate} \item if $A \to B$ is an unramified homomorphism of local rings, then $B^\wedge$ is a finite $A^\wedge$ module, \item if $A \to B$ is an unramified homomorphism of local rings and $\kappa(\mathfrak m_A) = \kappa(\mathfrak m_B)$, then $A^\wedge \to B^\wedge$ is surjective, \item if $A \to B$ is an unramified homomorphism of local rings and $\kappa(\mathfrak m_A)$ is separably closed, then $A^\wedge \to B^\wedge$ is surjective, \item if $A$ and $B$ are complete discrete valuation rings, then $A \to B$ is an unramified homomorphism of local rings if and only the uniformizer for $A$ maps to a uniformizer for $B$, and the residue field extension is finite separable (and $B$ is essentially of finite type over $A$). \end{enumerate} \end{lemma} \begin{proof} Part (1) is a special case of Algebra, Lemma \ref{algebra-lemma-finite-after-completion}. For part (2), note that the $\kappa(\mathfrak m_A)$-vector space $B^\wedge/\mathfrak m_{A^\wedge}B^\wedge$ is generated by $1$. Hence by Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK}) the map $A^\wedge \to B^\wedge$ is surjective. Part (3) is a special case of part (2). Part (4) is immediate from the definitions. \end{proof} \begin{lemma} \label{lemma-characterize-unramified-completions} Let $A$, $B$ be Noetherian local rings. Let $A \to B$ be a local homomorphism such that $B$ is essentially of finite type over $A$. The following are equivalent \begin{enumerate} \item $A \to B$ is an unramified homomorphism of local rings \item $A^\wedge \to B^\wedge$ is an unramified homomorphism of local rings, and \item $A^\wedge \to B^\wedge$ is unramified. \end{enumerate} \end{lemma} \begin{proof} The equivalence of (1) and (2) follows from the fact that $\mathfrak m_AA^\wedge$ is the maximal ideal of $A^\wedge$ (and similarly for $B$) and faithful flatness of $B \to B^\wedge$. For example if $A^\wedge \to B^\wedge$ is unramified, then $\mathfrak m_AB^\wedge = (\mathfrak m_AB)B^\wedge = \mathfrak m_BB^\wedge$ and hence $\mathfrak m_AB = \mathfrak m_B$. \medskip\noindent Assume the equivalent conditions (1) and (2). By Lemma \ref{lemma-unramified-completions} we see that $A^\wedge \to B^\wedge$ is finite. Hence $A^\wedge \to B^\wedge$ is of finite presentation, and by Algebra, Lemma \ref{algebra-lemma-characterize-unramified} we conclude that $A^\wedge \to B^\wedge$ is unramified at $\mathfrak m_{B^\wedge}$. Since $B^\wedge$ is local we conclude that $A^\wedge \to B^\wedge$ is unramified. \medskip\noindent Assume (3). By Algebra, Lemma \ref{algebra-lemma-unramified-at-prime} we conclude that $A^\wedge \to B^\wedge$ is an unramified homomorphism of local rings, i.e., (2) holds. \end{proof} \begin{definition} \label{definition-unramified-schemes} (See Morphisms, Definition \ref{morphisms-definition-unramified} for the definition in the general case.) Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be locally of finite type. Let $x \in X$. \begin{enumerate} \item We say $f$ is {\it unramified at $x$} if $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is an unramified homomorphism of local rings. \item The morphism $f : X \to Y$ is said to be {\it unramified} if it is unramified at all points of $X$. \end{enumerate} \end{definition} \noindent Let us prove that this definition agrees with the definition in the chapter on morphisms of schemes. This in particular guarantees that the set of points where a morphism is unramified is open. \begin{lemma} \label{lemma-unramified-definition} Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be locally of finite type. Let $x \in X$. The morphism $f$ is unramified at $x$ in the sense of Definition \ref{definition-unramified-schemes} if and only if it is unramified in the sense of Morphisms, Definition \ref{morphisms-definition-unramified}. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-characterize-unramified-Noetherian} and the definitions. \end{proof} \noindent Here are some results on unramified morphisms. The formulations as given in this list apply only to morphisms locally of finite type between locally Noetherian schemes. In each case we give a reference to the general result as proved earlier in the project, but in some cases one can prove the result more easily in the Noetherian case. Here is the list: \begin{enumerate} \item Unramifiedness is local on the source and the target in the Zariski topology. \item Unramified morphisms are stable under base change and composition. See Morphisms, Lemmas \ref{morphisms-lemma-base-change-unramified} and \ref{morphisms-lemma-composition-unramified}. \item Unramified morphisms of schemes are locally quasi-finite and quasi-compact unramified morphisms are quasi-finite. See Morphisms, Lemma \ref{morphisms-lemma-unramified-quasi-finite} \item Unramified morphisms have relative dimension $0$. See Morphisms, Definition \ref{morphisms-definition-relative-dimension-d} and Morphisms, Lemma \ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}. \item A morphism is unramified if and only if all its fibres are unramified. That is, unramifiedness can be checked on the scheme theoretic fibres. See Morphisms, Lemma \ref{morphisms-lemma-unramified-etale-fibres}. \item Let $X$ and $Y$ be unramified over a base scheme $S$. Any $S$-morphism from $X$ to $Y$ is unramified. See Morphisms, Lemma \ref{morphisms-lemma-unramified-permanence}. \end{enumerate} \section{Three other characterizations of unramified morphisms} \label{section-three-other} \noindent The following theorem gives three equivalent notions of being unramified at a point. See Morphisms, Lemma \ref{morphisms-lemma-unramified-at-point} for (part of) the statement for general schemes. \begin{theorem} \label{theorem-unramified-equivalence} Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism of schemes which is locally of finite type. Let $x$ be a point of $X$. The following are equivalent \begin{enumerate} \item $f$ is unramified at $x$, \item the stalk $\Omega_{X/Y, x}$ of the module of relative differentials at $x$ is trivial, \item there exist open neighbourhoods $U$ of $x$ and $V$ of $f(x)$, and a commutative diagram $$\xymatrix{ U \ar[rr]_i \ar[rd] & & \mathbf{A}^n_V \ar[ld] \\ & V }$$ where $i$ is a closed immersion defined by a quasi-coherent sheaf of ideals $\mathcal{I}$ such that the differentials $\text{d}g$ for $g \in \mathcal{I}_{i(x)}$ generate $\Omega_{\mathbf{A}^n_V/V, i(x)}$, and \item the diagonal $\Delta_{X/Y} : X \to X \times_Y X$ is a local isomorphism at $x$. \end{enumerate} \end{theorem} \begin{proof} The equivalence of (1) and (2) is proved in Morphisms, Lemma \ref{morphisms-lemma-unramified-at-point}. \medskip\noindent If $f$ is unramified at $x$, then $f$ is unramified in an open neighbourhood of $x$; this does not follow immediately from Definition \ref{definition-unramified-schemes} of this chapter but it does follow from Morphisms, Definition \ref{morphisms-definition-unramified} which we proved to be equivalent in Lemma \ref{lemma-unramified-definition}. Choose affine opens $V \subset Y$, $U \subset X$ with $f(U) \subset V$ and $x \in U$, such that $f$ is unramified on $U$, i.e., $f|_U : U \to V$ is unramified. By Morphisms, Lemma \ref{morphisms-lemma-diagonal-unramified-morphism} the morphism $U \to U \times_V U$ is an open immersion. This proves that (1) implies (4). \medskip\noindent If $\Delta_{X/Y}$ is a local isomorphism at $x$, then $\Omega_{X/Y, x} = 0$ by Morphisms, Lemma \ref{morphisms-lemma-differentials-diagonal}. Hence we see that (4) implies (2). At this point we know that (1), (2) and (4) are all equivalent. \medskip\noindent Assume (3). The assumption on the diagram combined with Morphisms, Lemma \ref{morphisms-lemma-differentials-relative-immersion} show that $\Omega_{U/V, x} = 0$. Since $\Omega_{U/V, x} = \Omega_{X/Y, x}$ we conclude (2) holds. \medskip\noindent Finally, assume that (2) holds. To prove (3) we may localize on $X$ and $Y$ and assume that $X$ and $Y$ are affine. Say $X = \Spec(B)$ and $Y = \Spec(A)$. The point $x \in X$ corresponds to a prime $\mathfrak q \subset B$. Our assumption is that $\Omega_{B/A, \mathfrak q} = 0$ (see Morphisms, Lemma \ref{morphisms-lemma-differentials-affine} for the relationship between differentials on schemes and modules of differentials in commutative algebra). Since $Y$ is locally Noetherian and $f$ locally of finite type we see that $A$ is Noetherian and $B \cong A[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$, see Properties, Lemma \ref{properties-lemma-locally-Noetherian} and Morphisms, Lemma \ref{morphisms-lemma-locally-finite-type-characterize}. In particular, $\Omega_{B/A}$ is a finite $B$-module. Hence we can find a single $g \in B$, $g \not \in \mathfrak q$ such that the principal localization $(\Omega_{B/A})_g$ is zero. Hence after replacing $B$ by $B_g$ we see that $\Omega_{B/A} = 0$ (formation of modules of differentials commutes with localization, see Algebra, Lemma \ref{algebra-lemma-differentials-localize}). This means that $\text{d}(f_j)$ generate the kernel of the canonical map $\Omega_{A[x_1, \ldots, x_n]/A} \otimes_A B \to \Omega_{B/A}$. Thus the surjection $A[x_1, \ldots, x_n] \to B$ of $A$-algebras gives the commutative diagram of (3), and the theorem is proved. \end{proof} \noindent How can we use this theorem? Well, here are a few remarks: \begin{enumerate} \item Suppose that $f : X \to Y$ and $g : Y \to Z$ are two morphisms locally of finite type between locally Noetherian schemes. There is a canonical short exact sequence $$f^*(\Omega_{Y/Z}) \to \Omega_{X/Z} \to \Omega_{X/Y} \to 0$$ see Morphisms, Lemma \ref{morphisms-lemma-triangle-differentials}. The theorem therefore implies that if $g \circ f$ is unramified, then so is $f$. This is Morphisms, Lemma \ref{morphisms-lemma-unramified-permanence}. \item Since $\Omega_{X/Y}$ is isomorphic to the conormal sheaf of the diagonal morphism (Morphisms, Lemma \ref{morphisms-lemma-differentials-diagonal}) we see that if $X \to Y$ is a monomorphism of locally Noetherian schemes and locally of finite type, then $X \to Y$ is unramified. In particular, open and closed immersions of locally Noetherian schemes are unramified. See Morphisms, Lemmas \ref{morphisms-lemma-open-immersion-unramified} and \ref{morphisms-lemma-closed-immersion-unramified}. \item The theorem also implies that the set of points where a morphism $f : X \to Y$ (locally of finite type of locally Noetherian schemes) is not unramified is the support of the coherent sheaf $\Omega_{X/Y}$. This allows one to give a scheme theoretic definition to the ramification locus''. \end{enumerate} \section{The functorial characterization of unramified morphisms} \label{section-functorial-unramified} \noindent In basic algebraic geometry we learn that some classes of morphisms can be characterized functorially, and that such descriptions are quite useful. Unramified morphisms too have such a characterization. \begin{theorem} \label{theorem-formally-unramified} Let $f : X \to S$ be a morphism of schemes. Assume $S$ is a locally Noetherian scheme, and $f$ is locally of finite type. Then the following are equivalent: \begin{enumerate} \item $f$ is unramified, \item the morphism $f$ is formally unramified: for any affine $S$-scheme $T$ and subscheme $T_0$ of $T$ defined by a square-zero ideal, the natural map $$\Hom_S(T, X) \longrightarrow \Hom_S(T_0, X)$$ is injective. \end{enumerate} \end{theorem} \begin{proof} See More on Morphisms, Lemma \ref{more-morphisms-lemma-unramified-formally-unramified} for a more general statement and proof. What follows is a sketch of the proof in the current case. \medskip\noindent Firstly, one checks both properties are local on the source and the target. This we may assume that $S$ and $X$ are affine. Say $X = \Spec(B)$ and $S = \Spec(R)$. Say $T = \Spec(C)$. Let $J$ be the square-zero ideal of $C$ with $T_0 = \Spec(C/J)$. Assume that we are given the diagram $$\xymatrix{ & B \ar[d]^\phi \ar[rd]^{\bar{\phi}} & \\ R \ar[r] \ar[ur] & C \ar[r] & C/J }$$ Secondly, one checks that the association $\phi' \mapsto \phi' - \phi$ gives a bijection between the set of liftings of $\bar{\phi}$ and the module $\text{Der}_R(B, J)$. Thus, we obtain the implication (1) $\Rightarrow$ (2) via the description of unramified morphisms having trivial module of differentials, see Theorem \ref{theorem-unramified-equivalence}. \medskip\noindent To obtain the reverse implication, consider the surjection $q : C = (B \otimes_R B)/I^2 \to B = C/J$ defined by the square zero ideal $J = I/I^2$ where $I$ is the kernel of the multiplication map $B \otimes_R B \to B$. We already have a lifting $B \to C$ defined by, say, $b \mapsto b \otimes 1$. Thus, by the same reasoning as above, we obtain a bijective correspondence between liftings of $\text{id} : B \to C/J$ and $\text{Der}_R(B, J)$. The hypothesis therefore implies that the latter module is trivial. But we know that $J \cong \Omega_{B/R}$. Thus, $B/R$ is unramified. \end{proof} \section{Topological properties of unramified morphisms} \label{section-topological-unramified} \noindent The first topological result that will be of utility to us is one which says that unramified and separated morphisms have nice'' sections. The material in this section does not require any Noetherian hypotheses. \begin{proposition} \label{proposition-properties-sections} Sections of unramified morphisms. \begin{enumerate} \item Any section of an unramified morphism is an open immersion. \item Any section of a separated morphism is a closed immersion. \item Any section of an unramified separated morphism is open and closed. \end{enumerate} \end{proposition} \begin{proof} Fix a base scheme $S$. If $f : X' \to X$ is any $S$-morphism, then the graph $\Gamma_f : X' \to X' \times_S X$ is obtained as the base change of the diagonal $\Delta_{X/S} : X \to X \times_S X$ via the projection $X' \times_S X \to X \times_S X$. If $g : X \to S$ is separated (resp. unramified) then the diagonal is a closed immersion (resp. open immersion) by Schemes, Definition \ref{schemes-definition-separated} (resp.\ Morphisms, Lemma \ref{morphisms-lemma-diagonal-unramified-morphism}). Hence so is the graph as a base change (by Schemes, Lemma \ref{schemes-lemma-base-change-immersion}). In the special case $X' = S$, we obtain (1), resp.\ (2). Part (3) follows on combining (1) and (2). \end{proof} \noindent We can now explicitly describe the sections of unramified morphisms. \begin{theorem} \label{theorem-sections-unramified-maps} Let $Y$ be a connected scheme. Let $f : X \to Y$ be unramified and separated. Every section of $f$ is an isomorphism onto a connected component. There exists a bijective correspondence $$\text{sections of }f \leftrightarrow \left\{ \begin{matrix} \text{connected components }X'\text{ of }X\text{ such that}\\ \text{the induced map }X' \to Y\text{ is an isomorphism} \end{matrix} \right\}$$ In particular, given $x \in X$ there is at most one section passing through $x$. \end{theorem} \begin{proof} Direct from Proposition \ref{proposition-properties-sections} part (3). \end{proof} \noindent The preceding theorem gives us some idea of the rigidity'' of unramified morphisms. Further indication is provided by the following proposition which, besides being intrinsically interesting, is also useful in the theory of the algebraic fundamental group (see \cite[Expos\'e V]{SGA1}). See also the more general Morphisms, Lemma \ref{morphisms-lemma-value-at-one-point}. \begin{proposition} \label{proposition-equality} Let $S$ is be a scheme. Let $\pi : X \to S$ be unramified and separated. Let $Y$ be an $S$-scheme and $y \in Y$ a point. Let $f, g : Y \to X$ be two $S$-morphisms. Assume \begin{enumerate} \item $Y$ is connected \item $x = f(y) = g(y)$, and \item the induced maps $f^\sharp, g^\sharp : \kappa(x) \to \kappa(y)$ on residue fields are equal. \end{enumerate} Then $f = g$. \end{proposition} \begin{proof} The maps $f, g : Y \to X$ define maps $f', g' : Y \to X_Y = Y \times_S X$ which are sections of the structure map $X_Y \to Y$. Note that $f = g$ if and only if $f' = g'$. The structure map $X_Y \to Y$ is the base change of $\pi$ and hence unramified and separated also (see Morphisms, Lemmas \ref{morphisms-lemma-base-change-unramified} and Schemes, Lemma \ref{schemes-lemma-separated-permanence}). Thus according to Theorem \ref{theorem-sections-unramified-maps} it suffices to prove that $f'$ and $g'$ pass through the same point of $X_Y$. And this is exactly what the hypotheses (2) and (3) guarantee, namely $f'(y) = g'(y) \in X_Y$. \end{proof} \begin{lemma} \label{lemma-finitely-many-maps-to-unramified} Let $S$ be a Noetherian scheme. Let $X \to S$ be a quasi-compact unramified morphism. Let $Y \to S$ be a morphism with $Y$ Noetherian. Then $\Mor_S(Y, X)$ is a finite set. \end{lemma} \begin{proof} Assume first $X \to S$ is separated (which is often the case in practice). Since $Y$ is Noetherian it has finitely many connected components. Thus we may assume $Y$ is connected. Choose a point $y \in Y$ with image $s \in S$. Since $X \to S$ is unramified and quasi-compact then fibre $X_s$ is finite, say $X_s = \{x_1, \ldots, x_n\}$ and $\kappa(s) \subset \kappa(x_i)$ is a finite field extension. See Morphisms, Lemma \ref{morphisms-lemma-unramified-quasi-finite}, \ref{morphisms-lemma-residue-field-quasi-finite}, and \ref{morphisms-lemma-quasi-finite}. For each $i$ there are at most finitely many $\kappa(s)$-algebra maps $\kappa(x_i) \to \kappa(y)$ (by elementary field theory). Thus $\Mor_S(Y, X)$ is finite by Proposition \ref{proposition-equality}. \medskip\noindent General case. There exists a nonempty open $U \subset X$ such that $X_U \to U$ is finite (in particular separated), see Morphisms, Lemma \ref{morphisms-lemma-generically-finite} (the lemma applies since we've already seen above that a quasi-compact unramified morphism is quasi-finite and since $X \to S$ is quasi-separated by Morphisms, Lemma \ref{morphisms-lemma-finite-type-Noetherian-quasi-separated}). Let $Z \subset S$ be the reduced closed subscheme supported on the complement of $U$. By Noetherian induction, we see that $\Mor_Z(Y_Z, X_Z)$ is finite (details omitted). By the result of the first paragraph the set $\Mor_U(Y_U, X_U)$ is finite. Thus it suffices to show that $$\Mor_S(Y, X) \longrightarrow \Mor_Z(Y_Z, X_Z) \times \Mor_U(Y_U, X_U)$$ is injective. This follows from the fact that the set of points where two morphisms $a, b : Y \to X$ agree is open in $Y$, due to the fact that $\Delta : X \to X \times_S X$ is open, see Morphisms, Lemma \ref{morphisms-lemma-diagonal-unramified-morphism}. \end{proof} \section{Universally injective, unramified morphisms} \label{section-universally-injective-unramified} \noindent Recall that a morphism of schemes $f : X \to Y$ is universally injective if any base change of $f$ is injective (on underlying topological spaces), see Morphisms, Definition \ref{morphisms-definition-universally-injective}. Universally injective and unramified morphisms can be characterized as follows. \begin{lemma} \label{lemma-universally-injective-unramified} Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item $f$ is unramified and a monomorphism, \item $f$ is unramified and universally injective, \item $f$ is locally of finite type and a monomorphism, \item $f$ is universally injective, locally of finite type, and formally unramified, \item $f$ is locally of finite type and $X_y$ is either empty or $X_y \to y$ is an isomorphism for all $y \in Y$. \end{enumerate} \end{lemma} \begin{proof} We have seen in More on Morphisms, Lemma \ref{more-morphisms-lemma-unramified-formally-unramified} that being formally unramified and locally of finite type is the same thing as being unramified. Hence (4) is equivalent to (2). A monomorphism is certainly universally injective and formally unramified hence (3) implies (4). It is clear that (1) implies (3). Finally, if (2) holds, then $\Delta : X \to X \times_S X$ is both an open immersion (Morphisms, Lemma \ref{morphisms-lemma-diagonal-unramified-morphism}) and surjective (Morphisms, Lemma \ref{morphisms-lemma-universally-injective}) hence an isomorphism, i.e., $f$ is a monomorphism. In this way we see that (2) implies (1). \medskip\noindent Condition (3) implies (5) because monomorphisms are preserved under base change (Schemes, Lemma \ref{schemes-lemma-base-change-monomorphism}) and because of the description of monomorphisms towards the spectra of fields in Schemes, Lemma \ref{schemes-lemma-mono-towards-spec-field}. Condition (5) implies (4) by Morphisms, Lemmas \ref{morphisms-lemma-universally-injective} and \ref{morphisms-lemma-unramified-etale-fibres}. \end{proof} \noindent This leads to the following useful characterization of closed immersions. \begin{lemma} \label{lemma-characterize-closed-immersion} Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item $f$ is a closed immersion, \item $f$ is a proper monomorphism, \item $f$ is proper, unramified, and universally injective, \item $f$ is universally closed, unramified, and a monomorphism, \item $f$ is universally closed, unramified, and universally injective, \item $f$ is universally closed, locally of finite type, and a monomorphism, \item $f$ is universally closed, universally injective, locally of finite type, and formally unramified. \end{enumerate} \end{lemma} \begin{proof} The equivalence of (4) -- (7) follows immediately from Lemma \ref{lemma-universally-injective-unramified}. \medskip\noindent Let $f : X \to S$ satisfy (6). Then $f$ is separated, see Schemes, Lemma \ref{schemes-lemma-monomorphism-separated} and has finite fibres. Hence More on Morphisms, Lemma \ref{more-morphisms-lemma-characterize-finite} shows $f$ is finite. Then Morphisms, Lemma \ref{morphisms-lemma-finite-monomorphism-closed} implies $f$ is a closed immersion, i.e., (1) holds. \medskip\noindent Note that (1) $\Rightarrow$ (2) because a closed immersion is proper and a monomorphism (Morphisms, Lemma \ref{morphisms-lemma-closed-immersion-proper} and Schemes, Lemma \ref{schemes-lemma-immersions-monomorphisms}). By Lemma \ref{lemma-universally-injective-unramified} we see that (2) implies (3). It is clear that (3) implies (5). \end{proof} \noindent Here is another result of a similar flavor. \begin{lemma} \label{lemma-finite-unramified-one-point} Let $\pi : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that \begin{enumerate} \item $\pi$ is finite, \item $\pi$ is unramified, \item $\pi^{-1}(\{s\}) = \{x\}$, and \item $\kappa(s) \subset \kappa(x)$ is purely inseparable\footnote{In view of condition (2) this is equivalent to $\kappa(s) = \kappa(x)$.}. \end{enumerate} Then there exists an open neighbourhood $U$ of $s$ such that $\pi|_{\pi^{-1}(U)} : \pi^{-1}(U) \to U$ is a closed immersion. \end{lemma} \begin{proof} The question is local on $S$. Hence we may assume that $S = \Spec(A)$. By definition of a finite morphism this implies $X = \Spec(B)$. Note that the ring map $\varphi : A \to B$ defining $\pi$ is a finite unramified ring map. Let $\mathfrak p \subset A$ be the prime corresponding to $s$. Let $\mathfrak q \subset B$ be the prime corresponding to $x$. By Conditions (2), (3) and (4) imply that $B_{\mathfrak q}/\mathfrak pB_{\mathfrak q} = \kappa(\mathfrak p)$. Algebra, Lemma \ref{algebra-lemma-unique-prime-over-localize-below} we have $B_{\mathfrak q} = B_{\mathfrak p}$ (note that a finite ring map satisfies going up, see Algebra, Section \ref{algebra-section-going-up}.) Hence we see that $B_{\mathfrak p}/\mathfrak pB_{\mathfrak p} = \kappa(\mathfrak p)$. As $B$ is a finite $A$-module we see from Nakayama's lemma (see Algebra, Lemma \ref{algebra-lemma-NAK}) that $B_{\mathfrak p} = \varphi(A_{\mathfrak p})$. Hence (using the finiteness of $B$ as an $A$-module again) there exists a $f \in A$, $f \not \in \mathfrak p$ such that $B_f = \varphi(A_f)$ as desired. \end{proof} \noindent The topological results presented above will be used to give a functorial characterization of \'etale morphisms similar to Theorem \ref{theorem-formally-unramified}. \section{Examples of unramified morphisms} \label{section-examples} \noindent Here are a few examples. \begin{example} \label{example-etale-field-extensions} Let $k$ be a field. Unramified quasi-compact morphisms $X \to \Spec(k)$ are affine. This is true because $X$ has dimension $0$ and is Noetherian, hence is a finite discrete set, and each point gives an affine open, so $X$ is a finite disjoint union of affines hence affine. Noether normalization forces $X$ to be the spectrum of a finite $k$-algebra $A$. This algebra is a product of finite separable field extensions of $k$. Thus, an unramified quasi-compact morphism to $\Spec(k)$ corresponds to a finite number of finite separable field extensions of $k$. In particular, an unramified morphism with a connected source and a one point target is forced to be a finite separable field extension. As we will see later, $X \to \Spec(k)$ is \'etale if and only if it is unramified. Thus, in this case at least, we obtain a very easy description of the \'etale topology of a scheme. Of course, the cohomology of this topology is another story. \end{example} \begin{example} \label{example-standard-etale} Property (3) in Theorem \ref{theorem-unramified-equivalence} gives us a canonical source of examples for unramified morphisms. Fix a ring $R$ and an integer $n$. Let $I = (g_1, \ldots, g_m)$ be an ideal in $R[x_1, \ldots, x_n]$. Let $\mathfrak q \subset R[x_1, \ldots, x_n]$ be a prime. Assume $I \subset \mathfrak q$ and that the matrix $$\left(\frac{\partial g_i}{\partial x_j}\right) \bmod \mathfrak q \quad\in\quad \text{Mat}(n \times m, \kappa(\mathfrak q))$$ has rank $n$. Then the morphism $f : Z = \Spec(R[x_1, \ldots, x_n]/I) \to \Spec(R)$ is unramified at the point $x \in Z \subset \mathbf{A}^n_R$ corresponding to $\mathfrak q$. Clearly we must have $m \geq n$. In the extreme case $m = n$, i.e., the differential of the map $\mathbf{A}^n_R \to \mathbf{A}^n_R$ defined by the $g_i$'s is an isomorphism of the tangent spaces, then $f$ is also flat $x$ and, hence, is an \'etale map (see Algebra, Definition \ref{algebra-definition-standard-smooth}, Lemma \ref{algebra-lemma-standard-smooth} and Example \ref{algebra-example-make-standard-smooth}). \end{example} \begin{example} \label{example-number-theory-etale} Fix an extension of number fields $L/K$ with rings of integers $\mathcal{O}_L$ and $\mathcal{O}_K$. The injection $K \to L$ defines a morphism $f : \Spec(\mathcal{O}_L) \to \Spec(\mathcal{O}_K)$. As discussed above, the points where $f$ is unramified in our sense correspond to the set of points where $f$ is unramified in the conventional sense. In the conventional sense, the locus of ramification in $\Spec(\mathcal{O}_L)$ can be defined by vanishing set of the different; this is an ideal in $\mathcal{O}_L$. In fact, the different is nothing but the annihilator of the module $\Omega_{\mathcal{O}_L/\mathcal{O}_K}$. Similarly, the discriminant is an ideal in $\mathcal{O}_K$, namely it is the norm of the different. The vanishing set of the discriminant is precisely the set of points of $K$ which ramify in $L$. Thus, denoting by $X$ the complement of the closed subset defined by the different in $\Spec(\mathcal{O}_L)$, we obtain a morphism $X \to \Spec(\mathcal{O}_L)$ which is unramified. Furthermore, this morphism is also flat, as any local homomorphism of discrete valuation rings is flat, and hence this morphism is actually \'etale. If $L/K$ is finite Galois, then denoting by $Y$ the complement of the closed subset defined by the discriminant in $\Spec(\mathcal{O}_K)$, we see that we get even a finite \'etale morphism $X \to Y$. Thus, this is an example of a finite \'etale covering. \end{example} \section{Flat morphisms} \label{section-flat-morphisms} \noindent This section simply exists to summarize the properties of flatness that will be useful to us. Thus, we will be content with stating the theorems precisely and giving references for the proofs. \medskip\noindent After briefly recalling the necessary facts about flat modules over Noetherian rings, we state a theorem of Grothendieck which gives sufficient conditions for hyperplane sections'' of certain modules to be flat. \begin{definition} \label{definition-flat-rings} Flatness of modules and rings. \begin{enumerate} \item A module $N$ over a ring $A$ is said to be {\it flat} if the functor $M \mapsto M \otimes_A N$ is exact. \item If this functor is also faithful, we say that $N$ is {\it faithfully flat} over $A$. \item A morphism of rings $f : A \to B$ is said to be {\it flat (resp. faithfully flat)} if the functor $M \mapsto M \otimes_A B$ is exact (resp. faithful and exact). \end{enumerate} \end{definition} \noindent Here is a list of facts with references to the algebra chapter. \begin{enumerate} \item Free and projective modules are flat. This is clear for free modules and follows for projective modules as they are direct summands of free modules and $\otimes$ commutes with direct sums. \item Flatness is a local property, that is, $M$ is flat over $A$ if and only if $M_{\mathfrak p}$ is flat over $A_{\mathfrak p}$ for all $\mathfrak p \in \Spec(A)$. See Algebra, Lemma \ref{algebra-lemma-flat-localization}. \item If $M$ is a flat $A$-module and $A \to B$ is a ring map, then $M \otimes_A B$ is a flat $B$-module. See Algebra, Lemma \ref{algebra-lemma-flat-base-change}. \item Finite flat modules over local rings are free. See Algebra, Lemma \ref{algebra-lemma-finite-flat-local}. \item If $f : A \to B$ is a morphism of arbitrary rings, $f$ is flat if and only if the induced maps $A_{f^{-1}(\mathfrak q)} \to B_{\mathfrak q}$ are flat for all $\mathfrak q \in \Spec(B)$. See Algebra, Lemma \ref{algebra-lemma-flat-localization} \item If $f : A \to B$ is a local homomorphism of local rings, $f$ is flat if and only if it is faithfully flat. See Algebra, Lemma \ref{algebra-lemma-local-flat-ff}. \item A map $A \to B$ of rings is faithfully flat if and only if it is flat and the induced map on spectra is surjective. See Algebra, Lemma \ref{algebra-lemma-ff-rings}. \item If $A$ is a noetherian local ring, the completion $A^\wedge$ is faithfully flat over $A$. See Algebra, Lemma \ref{algebra-lemma-completion-faithfully-flat}. \item Let $A$ be a Noetherian local ring and $M$ an $A$-module. Then $M$ is flat over $A$ if and only if $M \otimes_A A^\wedge$ is flat over $A^\wedge$. (Combine the previous statement with Algebra, Lemma \ref{algebra-lemma-flatness-descends}.) \end{enumerate} Before we move on to the geometric category, we present Grothendieck's theorem, which provides a convenient recipe for producing flat modules. \begin{theorem} \label{theorem-flatness-grothendieck} Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be a local homomorphism. If $M$ is a finite $B$-module that is flat as an $A$-module, and $t \in \mathfrak m_B$ is an element such that multiplication by $t$ is injective on $M/\mathfrak m_AM$, then $M/tM$ is also $A$-flat. \end{theorem} \begin{proof} See Algebra, Lemma \ref{algebra-lemma-mod-injective}. See also \cite[Section 20]{MatCA}. \end{proof} \begin{definition} \label{definition-flat-schemes} (See Morphisms, Definition \ref{morphisms-definition-flat}). Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item Let $x \in X$. We say $\mathcal{F}$ is {\it flat over $Y$ at $x \in X$} if $\mathcal{F}_x$ is a flat $\mathcal{O}_{Y, f(x)}$-module. This uses the map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ to think of $\mathcal{F}_x$ as a $\mathcal{O}_{Y, f(x)}$-module. \item Let $x \in X$. We say $f$ is {\it flat at $x \in X$} if $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is flat. \item We say $f$ is {\it flat} if it is flat at all points of $X$. \item A morphism $f : X \to Y$ that is flat and surjective is sometimes said to be {\it faithfully flat}. \end{enumerate} \end{definition} \noindent Once again, here is a list of results: \begin{enumerate} \item The property (of a morphism) of being flat is, by fiat, local in the Zariski topology on the source and the target. \item Open immersions are flat. (This is clear because it induces isomorphisms on local rings.) \item Flat morphisms are stable under base change and composition. Morphisms, Lemmas \ref{morphisms-lemma-base-change-flat} and \ref{morphisms-lemma-composition-flat}. \item If $f : X \to Y$ is flat, then the pullback functor $\QCoh(\mathcal{O}_Y) \to \QCoh(\mathcal{O}_X)$ is exact. This is immediate by looking at stalks. \item Let $f : X \to Y$ be a morphism of schemes, and assume $Y$ is quasi-compact and quasi-separated. In this case if the functor $f^*$ is exact then $f$ is flat. (Proof omitted. Hint: Use Properties, Lemma \ref{properties-lemma-extend-trivial} to see that $Y$ has enough'' ideal sheaves and use the characterization of flatness in Algebra, Lemma \ref{algebra-lemma-flat}.) \end{enumerate} \section{Topological properties of flat morphisms} \label{section-topological-flat} \noindent We recall'' below some openness properties that flat morphisms enjoy. \begin{theorem} \label{theorem-flat-open} Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism which is locally of finite type. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. The set of points in $X$ where $\mathcal{F}$ is flat over $Y$ is an open set. In particular the set of points where $f$ is flat is open in $X$. \end{theorem} \begin{proof} See More on Morphisms, Theorem \ref{more-morphisms-theorem-openness-flatness}. \end{proof} \begin{theorem} \label{theorem-flat-map-open} Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism which is flat and locally of finite type. Then $f$ is (universally) open. \end{theorem} \begin{proof} See Morphisms, Lemma \ref{morphisms-lemma-fppf-open}. \end{proof} \begin{theorem} \label{theorem-flat-is-quotient} A faithfully flat quasi-compact morphism is a quotient map for the Zariski topology. \end{theorem} \begin{proof} See Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology}. \end{proof} \noindent An important reason to study flat morphisms is that they provide the adequate framework for capturing the notion of a family of schemes parametrized by the points of another scheme. Naively one may think that any morphism $f : X \to S$ should be thought of as a family parametrized by the points of $S$. However, without a flatness restriction on $f$, really bizarre things can happen in this so-called family. For instance, we aren't guaranteed that relative dimension (dimension of the fibres) is constant in a family. Other numerical invariants, such as the Hilbert polynomial, too may change from fibre to fibre. Flatness prevents such things from happening and, therefore, provides some continuity'' to the fibres. \section{\'Etale morphisms} \label{section-etale-morphisms} \noindent In this section, we will define \'etale morphisms and prove a number of important properties about them. The most important one, no doubt, is the functorial characterization presented in Theorem \ref{theorem-formally-etale}. Following this, we will also discuss a few properties of rings which are insensitive to an \'etale extension (properties which hold for a ring if and only if they hold for all its \'etale extensions) to motivate the basic tenet of \'etale cohomology -- \'etale morphisms are the algebraic analogue of local isomorphisms. \medskip\noindent As the title suggests, we will define the class of \'etale morphisms -- the class of morphisms (whose surjective families) we shall deem to be coverings in the category of schemes over a base scheme $S$ in order to define the \'etale site $S_\etale$. Intuitively, an \'etale morphism is supposed to capture the idea of a covering space and, therefore, should be close to a local isomorphism. If we're working with varieties over algebraically closed fields, this last statement can be made into a definition provided we replace local isomorphism'' with formal local isomorphism'' (isomorphism after completion). One can then give a definition over any base field by asking that the base change to the algebraic closure be \'etale (in the aforementioned sense). But, rather than proceeding via such aesthetically displeasing constructions, we will adopt a cleaner, albeit slightly more abstract, algebraic approach. \medskip\noindent We first define \'etale homomorphisms of local rings'' for Noetherian local rings. We cannot use the term \'etale'', as there already is a notion of an \'etale ring map (Algebra, Section \ref{algebra-section-etale}) and it is different. \begin{definition} \label{definition-etale-ring} Let $A$, $B$ be Noetherian local rings. A local homomorphism $f : A \to B$ is said to be a {\it \'etale homomorphism of local rings} if it is flat and an unramified homomorphism of local rings (please see Definition \ref{definition-unramified-rings}). \end{definition} \noindent This is the local version of the definition of an \'etale ring map in Algebra, Section \ref{algebra-section-etale}. The exact definition given in that section is that it is a smooth ring map of relative dimension $0$. It is shown (in Algebra, Lemma \ref{algebra-lemma-etale-standard-smooth}) that an \'etale $R$-algebra $S$ always has a presentation $$S = R[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$$ such that $$g = \det \left( \begin{matrix} \partial f_1/\partial x_1 & \partial f_2/\partial x_1 & \ldots & \partial f_n/\partial x_1 \\ \partial f_1/\partial x_2 & \partial f_2/\partial x_2 & \ldots & \partial f_n/\partial x_2 \\ \ldots & \ldots & \ldots & \ldots \\ \partial f_1/\partial x_n & \partial f_2/\partial x_n & \ldots & \partial f_n/\partial x_n \end{matrix} \right)$$ maps to an invertible element in $S$. The following two lemmas link the two notions. \begin{lemma} \label{lemma-characterize-etale-Noetherian} Let $A \to B$ be of finite type with $A$ a Noetherian ring. Let $\mathfrak q$ be a prime of $B$ lying over $\mathfrak p \subset A$. Then $A \to B$ is \'etale at $\mathfrak q$ if and only if $A_{\mathfrak p} \to B_{\mathfrak q}$ is an \'etale homomorphism of local rings. \end{lemma} \begin{proof} See Algebra, Lemmas \ref{algebra-lemma-etale} (flatness of \'etale maps), \ref{algebra-lemma-etale-at-prime} (\'etale maps are unramified) and \ref{algebra-lemma-characterize-etale} (flat and unramified maps are \'etale). \end{proof} \begin{lemma} \label{lemma-characterize-etale-completions} Let $A$, $B$ be Noetherian local rings. Let $A \to B$ be a local homomorphism such that $B$ is essentially of finite type over $A$. The following are equivalent \begin{enumerate} \item $A \to B$ is an \'etale homomorphism of local rings \item $A^\wedge \to B^\wedge$ is an \'etale homomorphism of local rings, and \item $A^\wedge \to B^\wedge$ is \'etale. \end{enumerate} Moreover, in this case $B^\wedge \cong (A^\wedge)^{\oplus n}$ as $A^\wedge$-modules for some $n \geq 1$. \end{lemma} \begin{proof} To see the equivalences of (1), (2) and (3), as we have the corresponding results for unramified ring maps (Lemma \ref{lemma-characterize-unramified-completions}) it suffices to prove that $A \to B$ is flat if and only if $A^\wedge \to B^\wedge$ is flat. This is clear from our lists of properties of flat maps since the ring maps $A \to A^\wedge$ and $B \to B^\wedge$ are faithfully flat. For the final statement, by Lemma \ref{lemma-unramified-completions} we see that $B^\wedge$ is a finite flat $A^\wedge$ module. Hence it is finite free by our list of properties on flat modules in Section \ref{section-flat-morphisms}. \end{proof} \noindent The integer $n$ which occurs in the lemma above is nothing other than the degree $[\kappa(\mathfrak m_B) : \kappa(\mathfrak m_A)]$ of the residue field extension. In particular, if $\kappa(\mathfrak m_A)$ is separably closed, we see that $A^\wedge \to B^\wedge$ is an isomorphism, which vindicates our earlier claims. \begin{definition} \label{definition-etale-schemes-1} (See Morphisms, Definition \ref{morphisms-definition-etale}.) Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism of schemes which is locally of finite type. \begin{enumerate} \item Let $x \in X$. We say $f$ is {\it \'etale at $x \in X$} if $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is an \'etale homomorphism of local rings. \item The morphism is said to be {\it \'etale} if it is \'etale at all its points. \end{enumerate} \end{definition} \noindent Let us prove that this definition agrees with the definition in the chapter on morphisms of schemes. This in particular guarantees that the set of points where a morphism is \'etale is open. \begin{lemma} \label{lemma-etale-definition} Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be locally of finite type. Let $x \in X$. The morphism $f$ is \'etale at $x$ in the sense of Definition \ref{definition-etale-schemes-1} if and only if it is \'etale at $x$ in the sense of Morphisms, Definition \ref{morphisms-definition-etale}. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-characterize-etale-Noetherian} and the definitions. \end{proof} \noindent Here are some results on \'etale morphisms. The formulations as given in this list apply only to morphisms locally of finite type between locally Noetherian schemes. In each case we give a reference to the general result as proved earlier in the project, but in some cases one can prove the result more easily in the Noetherian case. Here is the list: \begin{enumerate} \item An \'etale morphism is unramified. (Clear from our definitions.) \item \'Etaleness is local on the source and the target in the Zariski topology. \item \'Etale morphisms are stable under base change and composition. See Morphisms, Lemmas \ref{morphisms-lemma-base-change-etale} and \ref{morphisms-lemma-composition-etale}. \item \'Etale morphisms of schemes are locally quasi-finite and quasi-compact \'etale morphisms are quasi-finite. (This is true because it holds for unramified morphisms as seen earlier.) \item \'Etale morphisms have relative dimension $0$. See Morphisms, Definition \ref{morphisms-definition-relative-dimension-d} and Morphisms, Lemma \ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}. \item A morphism is \'etale if and only if it is flat and all its fibres are \'etale. See Morphisms, Lemma \ref{morphisms-lemma-etale-flat-etale-fibres}. \item \'Etale morphisms are open. This is true because an \'etale morphism is flat, and Theorem \ref{theorem-flat-map-open}. \item Let $X$ and $Y$ be \'etale over a base scheme $S$. Any $S$-morphism from $X$ to $Y$ is \'etale. See Morphisms, Lemma \ref{morphisms-lemma-etale-permanence}. \end{enumerate} \section{The structure theorem} \label{section-structure-etale-map} \noindent We present a theorem which describes the local structure of \'etale and unramified morphisms. Besides its obvious independent importance, this theorem also allows us to make the transition to another definition of \'etale morphisms that captures the geometric intuition better than the one we've used so far. \medskip\noindent To state it we need the notion of a {\it standard \'etale ring map}, see Algebra, Definition \ref{algebra-definition-standard-etale}. Namely, suppose that $R$ is a ring and $f, g \in R[t]$ are polynomials such that \begin{enumerate} \item[(a)] $f$ is a monic polynomial, and \item[(b)] $f' = \text{d}f/\text{d}t$ is invertible in the localization $R[t]_g/(f)$. \end{enumerate} Then the map $$R \longrightarrow R[t]_g/(f) = R[t, 1/g]/(f)$$ is a standard \'etale algebra, and any standard \'etale algebra is isomorphic to one of these. It is a pleasant exercise to prove that such a ring map is flat, and unramified and hence \'etale (as expected of course). A special case of a standard \'etale ring map is any ring map $$R \longrightarrow R[t]_{f'}/(f) = R[t, 1/f']/(f)$$ with $f$ a monic polynomial, and any standard \'etale algebra is (isomorphic to) a principal localization of one of these. \begin{theorem} \label{theorem-structure-etale} Let $f : A \to B$ be an \'etale homomorphism of local rings. Then there exist $f, g \in A[t]$ such that \begin{enumerate} \item $B' = A[t]_g/(f)$ is standard \'etale -- see (a) and (b) above, and \item $B$ is isomorphic to a localization of $B'$ at a prime. \end{enumerate} \end{theorem} \begin{proof} Write $B = B'_{\mathfrak q}$ for some finite type $A$-algebra $B'$ (we can do this because $B$ is essentially of finite type over $A$). By Lemma \ref{lemma-characterize-etale-Noetherian} we see that $A \to B'$ is \'etale at $\mathfrak q$. Hence we may apply Algebra, Proposition \ref{algebra-proposition-etale-locally-standard} to see that a principal localization of $B'$ is standard \'etale. \end{proof} \noindent Here is the version for unramified homomorphisms of local rings. \begin{theorem} \label{theorem-structure-unramified} Let $f : A \to B$ be an unramified morphism of local rings. Then there exist $f, g \in A[t]$ such that \begin{enumerate} \item $B' = A[t]_g/(f)$ is standard \'etale -- see (a) and (b) above, and \item $B$ is isomorphic to a quotient of a localization of $B'$ at a prime. \end{enumerate} \end{theorem} \begin{proof} Write $B = B'_{\mathfrak q}$ for some finite type $A$-algebra $B'$ (we can do this because $B$ is essentially of finite type over $A$). By Lemma \ref{lemma-characterize-unramified-Noetherian} we see that $A \to B'$ is unramified at $\mathfrak q$. Hence we may apply Algebra, Proposition \ref{algebra-proposition-unramified-locally-standard} to see that a principal localization of $B'$ is a quotient of a standard \'etale $A$-algebra. \end{proof} \noindent Via standard lifting arguments, one then obtains the following geometric statement which will be of essential use to us. \begin{theorem} \label{theorem-geometric-structure} Let $\varphi : X \to Y$ be a morphism of schemes. Let $x \in X$. Let $V \subset Y$ be an affine open neighbourhood of $\varphi(x)$. If $\varphi$ is \'etale at $x$, then there exist exists an affine open $U \subset X$ with $x \in U$ and $\varphi(U) \subset V$ such that we have the following diagram $$\xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-j & \Spec(R[t]_{f'}/(f)) \ar[d] \\ Y & V \ar[l] \ar@{=}[r] & \Spec(R) }$$ where $j$ is an open immersion, and $f \in R[t]$ is monic. \end{theorem} \begin{proof} This is equivalent to Morphisms, Lemma \ref{morphisms-lemma-etale-locally-standard-etale} although the statements differ slightly. See also, Varieties, Lemma \ref{varieties-lemma-geometric-structure-unramified} for a variant for unramified morphisms. \end{proof} \section{\'Etale and smooth morphisms} \label{section-etale-smooth} \noindent An \'etale morphism is smooth of relative dimension zero. The projection $\mathbf{A}^n_S \to S$ is a standard example of a smooth morphism of relative dimension $n$. It turns out that any smooth morphism is \'etale locally of this form. Here is the precise statement. \begin{theorem} \label{theorem-smooth-etale-over-n-space} Let $\varphi : X \to Y$ be a morphism of schemes. Let $x \in X$. If $\varphi$ is smooth at $x$, then there exist exist and integer $n \geq 0$ and affine opens $V \subset Y$ and $U \subset X$ with $x \in U$ and $\varphi(U) \subset V$ such that there exists a commutative diagram $$\xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-\pi & \mathbf{A}^n_R \ar[d] \ar@{=}[r] & \Spec(R[x_1, \ldots, x_n]) \ar[dl] \\ Y & V \ar[l] \ar@{=}[r] & \Spec(R) }$$ where $\pi$ is \'etale. \end{theorem} \begin{proof} See Morphisms, Lemma \ref{morphisms-lemma-smooth-etale-over-affine-space}. \end{proof} \section{Topological properties of \'etale morphisms} \label{section-topological-etale} \noindent We present a few of the topological properties of \'etale and unramified morphisms. First, we give what Grothendieck calls the {\it fundamental property of \'etale morphisms}, see \cite[Expos\'e I.5]{SGA1}. \begin{theorem} \label{theorem-etale-radicial-open} Let $f : X \to Y$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item $f$ is an open immersion, \item $f$ is universally injective and \'etale, and \item $f$ is a flat monomorphism, locally of finite presentation. \end{enumerate} \end{theorem} \begin{proof} An open immersion is universally injective since any base change of an open immersion is an open immersion. Moreover, it is \'etale by Morphisms, Lemma \ref{morphisms-lemma-open-immersion-etale}. Hence (1) implies (2). \medskip\noindent Assume $f$ is universally injective and \'etale. Since $f$ is \'etale it is flat and locally of finite presentation, see Morphisms, Lemmas \ref{morphisms-lemma-etale-flat} and \ref{morphisms-lemma-etale-locally-finite-presentation}. By Lemma \ref{lemma-universally-injective-unramified} we see that $f$ is a monomorphism. Hence (2) implies (3). \medskip\noindent Assume $f$ is flat, locally of finite presentation, and a monomorphism. Then $f$ is open, see Morphisms, Lemma \ref{morphisms-lemma-fppf-open}. Thus we may replace $Y$ by $f(X)$ and we may assume $f$ is surjective. Then $f$ is open and bijective hence a homeomorphism. Hence $f$ is quasi-compact. Hence Descent, Lemma \ref{descent-lemma-flat-surjective-quasi-compact-monomorphism-isomorphism} shows that $f$ is an isomorphism and we win. \end{proof} \noindent Here is another result of a similar flavor. \begin{lemma} \label{lemma-finite-etale-one-point} Let $\pi : X \to S$ be a morphism of schemes. Let $s \in S$. Assume that \begin{enumerate} \item $\pi$ is finite, \item $\pi$ is \'etale, \item $\pi^{-1}(\{s\}) = \{x\}$, and \item $\kappa(s) \subset \kappa(x)$ is purely inseparable\footnote{In view of condition (2) this is equivalent to $\kappa(s) = \kappa(x)$.}. \end{enumerate} Then there exists an open neighbourhood $U$ of $s$ such that $\pi|_{\pi^{-1}(U)} : \pi^{-1}(U) \to U$ is an isomorphism. \end{lemma} \begin{proof} By Lemma \ref{lemma-finite-unramified-one-point} there exists an open neighbourhood $U$ of $s$ such that $\pi|_{\pi^{-1}(U)} : \pi^{-1}(U) \to U$ is a closed immersion. But a morphism which is \'etale and a closed immersion is an open immersion (for example by Theorem \ref{theorem-etale-radicial-open}). Hence after shrinking $U$ we obtain an isomorphism. \end{proof} \section{Topological invariance of the \'etale topology} \label{section-topological-invariance} \noindent Next, we present an extremely crucial theorem which, roughly speaking, says that \'etaleness is a topological property. \begin{theorem} \label{theorem-etale-topological} Let $X$ and $Y$ be two schemes over a base scheme $S$. Let $S_0$ be a closed subscheme of $S$ with the same underlying topological space (for example if the ideal sheaf of $S_0$ in $S$ has square zero). Denote $X_0$ (resp.\ $Y_0$) the base change $S_0 \times_S X$ (resp.\ $S_0 \times_S Y$). If $X$ is \'etale over $S$, then the map $$\Mor_S(Y, X) \longrightarrow \Mor_{S_0}(Y_0, X_0)$$ is bijective. \end{theorem} \begin{proof} After base changing via $Y \to S$, we may assume that $Y = S$. In this case the theorem states that any $S$-morphism $\sigma_0 : S_0 \to X$ actually factors uniquely through a section $S \to X$ of the \'etale structure morphism $f : X \to S$. \medskip\noindent Uniqueness. Suppose we have two sections $\sigma, \sigma'$ through which $\sigma_0$ factors. Because $X \to S$ is \'etale we see that $\Delta : X \to X \times_S X$ is an open immersion (Morphisms, Lemma \ref{morphisms-lemma-diagonal-unramified-morphism}). The morphism $(\sigma, \sigma') : S \to X \times_S X$ factors through this open because for any $s \in S$ we have $(\sigma, \sigma')(s) = (\sigma_0(s), \sigma_0(s))$. Thus $\sigma = \sigma'$. \medskip\noindent To prove existence we first reduce to the affine case (we suggest the reader skip this step). Let $X = \bigcup X_i$ be an affine open covering such that each $X_i$ maps into an affine open $S_i$ of $S$. For every $s \in S$ we can choose an $i$ such that $\sigma_0(s) \in X_i$. Choose an affine open neighbourhood $U \subset S_i$ of $s$ such that $\sigma_0(U_0) \subset X_{i, 0}$. Note that $X' = X_i \times_S U = X_i \times_{S_i} U$ is affine. If we can lift $\sigma_0|_{U_0} : U_0 \to X'_0$ to $U \to X'$, then by uniqueness these local lifts will glue to a global morphism $S \to X$. Thus we may assume $S$ and $X$ are affine. \medskip\noindent Existence when $S$ and $X$ are affine. Write $S = \Spec(A)$ and $X = \Spec(B)$. Then $A \to B$ is \'etale and in particular smooth (of relative dimension $0$). As $|S_0| = |S|$ we see that $S_0 = \Spec(A/I)$ with $I \subset A$ locally nilpotent. Thus existence follows from Algebra, Lemma \ref{algebra-lemma-smooth-strong-lift}. \end{proof} \noindent From the proof of preceeding theorem, we also obtain one direction of the promised functorial characterization of \'etale morphisms. The following theorem will be strengthened in \'Etale Cohomology, Theorem \ref{etale-cohomology-theorem-topological-invariance}. \begin{theorem}[Une equivalence remarquable de cat\'egories] \label{theorem-remarkable-equivalence} Let $S$ be a scheme. Let $S_0 \subset S$ be a closed subscheme with the same underlying topological space (for example if the ideal sheaf of $S_0$ in $S$ has square zero). The functor $$X \longmapsto X_0 = S_0 \times_S X$$ defines an equivalence of categories $$\{ \text{schemes }X\text{ \'etale over }S \} \leftrightarrow \{ \text{schemes }X_0\text{ \'etale over }S_0 \}$$ \end{theorem} \begin{proof} By Theorem \ref{theorem-etale-topological} we see that this functor is fully faithful. It remains to show that the functor is essentially surjective. Let $Y \to S_0$ be an \'etale morphism of schemes. \medskip\noindent Suppose that the result holds if $S$ and $Y$ are affine. In that case, we choose an affine open covering $Y = \bigcup V_j$ such that each $V_j$ maps into an affine open of $S$. By assumption (affine case) we can find \'etale morphisms $W_j \to S$ such that $W_{j, 0} \cong V_j$ (as schemes over $S_0$). Let $W_{j, j'} \subset W_j$ be the open subscheme whose underlying topological space corresponds to $V_j \cap V_{j'}$. Because we have isomorphisms $$W_{j, j', 0} \cong V_j \cap V_{j'} \cong W_{j', j, 0}$$ as schemes over $S_0$ we see by fully faithfulness that we obtain isomorphisms $\theta_{j, j'} : W_{j, j'} \to W_{j', j}$ of schemes over $S$. We omit the verification that these isomorphisms satisfy the cocycle condition of Schemes, Section \ref{schemes-section-glueing-schemes}. Applying Schemes, Lemma \ref{schemes-lemma-glue-schemes} we obtain a scheme $X \to S$ by glueing the schemes $W_j$ along the identifications $\theta_{j, j'}$. It is clear that $X \to S$ is \'etale and $X_0 \cong Y$ by construction. \medskip\noindent Thus it suffices to show the lemma in case $S$ and $Y$ are affine. Say $S = \Spec(R)$ and $S_0 = \Spec(R/I)$ with $I$ locally nilpotent. By Algebra, Lemma \ref{algebra-lemma-etale-standard-smooth} we know that $Y$ is the spectrum of a ring $\overline{A}$ with $$\overline{A} = (R/I)[x_1, \ldots, x_n]/(\overline{f}_1, \ldots, \overline{f}_n)$$ such that $$\overline{g} = \det \left( \begin{matrix} \partial \overline{f}_1/\partial x_1 & \partial \overline{f}_2/\partial x_1 & \ldots & \partial \overline{f}_n/\partial x_1 \\ \partial \overline{f}_1/\partial x_2 & \partial \overline{f}_2/\partial x_2 & \ldots & \partial \overline{f}_n/\partial x_2 \\ \ldots & \ldots & \ldots & \ldots \\ \partial \overline{f}_1/\partial x_n & \partial \overline{f}_2/\partial x_n & \ldots & \partial \overline{f}_n/\partial x_n \end{matrix} \right)$$ maps to an invertible element in $\overline{A}$. Choose any lifts $f_i \in R[x_1, \ldots, x_n]$. Set $$A = R[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$$ Since $I$ is locally nilpotent the ideal $IA$ is locally nilpotent (Algebra, Lemma \ref{algebra-lemma-locally-nilpotent}). Observe that $\overline{A} = A/IA$. It follows that the determinant of the matrix of partials of the $f_i$ is invertible in the algebra $A$ by Algebra, Lemma \ref{algebra-lemma-locally-nilpotent-unit}. Hence $R \to A$ is \'etale and the proof is complete. \end{proof} \section{The functorial characterization} \label{section-functorial-etale} \noindent We finally present the promised functorial characterization. Thus there are four ways to think about \'etale morphisms of schemes: \begin{enumerate} \item as a smooth morphism of relative dimension $0$, \item as locally finitely presented, flat, and unramified morphisms, \item using the structure theorem, and \item using the functorial characterization. \end{enumerate} \begin{theorem} \label{theorem-formally-etale} Let $f : X \to S$ be a morphism that is locally of finite presentation. The following are equivalent \begin{enumerate} \item $f$ is \'etale, \item for all affine $S$-schemes $Y$, and closed subschemes $Y_0 \subset Y$ defined by square-zero ideals, the natural map $$\Mor_S(Y, X) \longrightarrow \Mor_S(Y_0, X)$$ is bijective. \end{enumerate} \end{theorem} \begin{proof} This is More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-formally-etale}. \end{proof} \noindent This characterization says that solutions to the equations defining $X$ can be lifted uniquely through nilpotent thickenings. \section{\'Etale local structure of unramified morphisms} \label{section-unramified-etale-local} \noindent In the chapter More on Morphisms, Section \ref{more-morphisms-section-etale-localization} the reader can find some results on the \'etale local structure of quasi-finite morphisms. In this section we want to combine this with the topological properties of unramified morphisms we have seen in this chapter. The basic overall picture to keep in mind is $$\xymatrix{ V \ar[r] \ar[dr] & X_U \ar[d] \ar[r] & X \ar[d]^f \\ & U \ar[r] & S }$$ see More on Morphisms, Equation (\ref{more-morphisms-equation-basic-diagram}). We start with a very general case. \begin{lemma} \label{lemma-unramified-etale-local} Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots, x_n \in X$ be points having the same image $s$ in $S$. Assume $f$ is unramified at each $x_i$. Then there exists an \'etale neighbourhood $(U, u) \to (S, s)$ and opens $V_{i, j} \subset X_U$, $i = 1, \ldots, n$, $j = 1, \ldots, m_i$ such that \begin{enumerate} \item $V_{i, j} \to U$ is a closed immersion passing through $u$, \item $u$ is not in the image of $V_{i, j} \cap V_{i', j'}$ unless $i = i'$ and $j = j'$, and \item any point of $(X_U)_u$ mapping to $x_i$ is in some $V_{i, j}$. \end{enumerate} \end{lemma} \begin{proof} By Morphisms, Definition \ref{morphisms-definition-unramified} there exists an open neighbourhood of each $x_i$ which is locally of finite type over $S$. Replacing $X$ by an open neighbourhood of $\{x_1, \ldots, x_n\}$ we may assume $f$ is locally of finite type. Apply More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-makes-quasi-finite-finite-multiple-points-var} to get the \'etale neighbourhood $(U, u)$ and the opens $V_{i, j}$ finite over $U$. By Lemma \ref{lemma-finite-unramified-one-point} after possibly shrinking $U$ we get that $V_{i, j} \to U$ is a closed immersion. \end{proof} \begin{lemma} \label{lemma-unramified-etale-local-technical} Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots, x_n \in X$ be points having the same image $s$ in $S$. Assume $f$ is separated and $f$ is unramified at each $x_i$. Then there exists an \'etale neighbourhood $(U, u) \to (S, s)$ and a disjoint union decomposition $$X_U = W \amalg \coprod\nolimits_{i, j} V_{i, j}$$ such that \begin{enumerate} \item $V_{i, j} \to U$ is a closed immersion passing through $u$, \item the fibre $W_u$ contains no point mapping to any $x_i$. \end{enumerate} In particular, if $f^{-1}(\{s\}) = \{x_1, \ldots, x_n\}$, then the fibre $W_u$ is empty. \end{lemma} \begin{proof} Apply Lemma \ref{lemma-unramified-etale-local}. We may assume $U$ is affine, so $X_U$ is separated. Then $V_{i, j} \to X_U$ is a closed map, see Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed}. Suppose $(i, j) \not = (i', j')$. Then $V_{i, j} \cap V_{i', j'}$ is closed in $V_{i, j}$ and its image in $U$ does not contain $u$. Hence after shrinking $U$ we may assume that $V_{i, j} \cap V_{i', j'} = \emptyset$. Moreover, $\bigcup V_{i, j}$ is a closed and open subscheme of $X_U$ and hence has an open and closed complement $W$. This finishes the proof. \end{proof} \noindent The following lemma is in some sense much weaker than the preceding one but it may be useful to state it explicitly here. It says that a finite unramified morphism is \'etale locally on the base a closed immersion. \begin{lemma} \label{lemma-finite-unramified-etale-local} Let $f : X \to S$ be a finite unramified morphism of schemes. Let $s \in S$. There exists an \'etale neighbourhood $(U, u) \to (S, s)$ and a disjoint union decomposition $$X_U = \coprod\nolimits_j V_j$$ such that each $V_j \to U$ is a closed immersion. \end{lemma} \begin{proof} Since $X \to S$ is finite the fibre over $S$ is a finite set $\{x_1, \ldots, x_n\}$ of points of $X$. Apply Lemma \ref{lemma-unramified-etale-local-technical} to this set (a finite morphism is separated, see Morphisms, Section \ref{morphisms-section-integral}). The image of $W$ in $U$ is a closed subset (as $X_U \to U$ is finite, hence proper) which does not contain $u$. After removing this from $U$ we see that $W = \emptyset$ as desired. \end{proof} \section{\'Etale local structure of \'etale morphisms} \label{section-etale-local-etale} \noindent This is a bit silly, but perhaps helps form intuition about \'etale morphisms. We simply copy over the results of Section \ref{section-unramified-etale-local} and change closed immersion'' into isomorphism''. \begin{lemma} \label{lemma-etale-etale-local} Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots, x_n \in X$ be points having the same image $s$ in $S$. Assume $f$ is \'etale at each $x_i$. Then there exists an \'etale neighbourhood $(U, u) \to (S, s)$ and opens $V_{i, j} \subset X_U$, $i = 1, \ldots, n$, $j = 1, \ldots, m_i$ such that \begin{enumerate} \item $V_{i, j} \to U$ is an isomorphism, \item $u$ is not in the image of $V_{i, j} \cap V_{i', j'}$ unless $i = i'$ and $j = j'$, and \item any point of $(X_U)_u$ mapping to $x_i$ is in some $V_{i, j}$. \end{enumerate} \end{lemma} \begin{proof} An \'etale morphism is unramified, hence we may apply Lemma \ref{lemma-unramified-etale-local}. Now $V_{i, j} \to U$ is a closed immersion and \'etale. Hence it is an open immersion, for example by Theorem \ref{theorem-etale-radicial-open}. Replace $U$ by the intersection of the images of $V_{i, j} \to U$ to get the lemma. \end{proof} \begin{lemma} \label{lemma-etale-etale-local-technical} Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots, x_n \in X$ be points having the same image $s$ in $S$. Assume $f$ is separated and $f$ is \'etale at each $x_i$. Then there exists an \'etale neighbourhood $(U, u) \to (S, s)$ and a disjoint union decomposition $$X_U = W \amalg \coprod\nolimits_{i, j} V_{i, j}$$ such that \begin{enumerate} \item $V_{i, j} \to U$ is an isomorphism, \item the fibre $W_u$ contains no point mapping to any $x_i$. \end{enumerate} In particular, if $f^{-1}(\{s\}) = \{x_1, \ldots, x_n\}$, then the fibre $W_u$ is empty. \end{lemma} \begin{proof} An \'etale morphism is unramified, hence we may apply Lemma \ref{lemma-unramified-etale-local-technical}. As in the proof of Lemma \ref{lemma-etale-etale-local} the morphisms $V_{i, j} \to U$ are open immersions and we win after replacing $U$ by the intersection of their images. \end{proof} \noindent The following lemma is in some sense much weaker than the preceding one but it may be useful to state it explicitly here. It says that a finite \'etale morphism is \'etale locally on the base a topological covering space'', i.e., a finite product of copies of the base. \begin{lemma} \label{lemma-finite-etale-etale-local} Let $f : X \to S$ be a finite \'etale morphism of schemes. Let $s \in S$. There exists an \'etale neighbourhood $(U, u) \to (S, s)$ and a disjoint union decomposition $$X_U = \coprod\nolimits_j V_j$$ such that each $V_j \to U$ is an isomorphism. \end{lemma} \begin{proof} An \'etale morphism is unramified, hence we may apply Lemma \ref{lemma-finite-unramified-etale-local}. As in the proof of Lemma \ref{lemma-etale-etale-local} we see that $V_{i, j} \to U$ is an open immersion and we win after replacing $U$ by the intersection of their images. \end{proof} \section{Permanence properties} \label{section-properties-permanence} \noindent In what follows, we present a few permanence'' properties of \'etale homomorphisms of Noetherian local rings (as defined in Definition \ref{definition-etale-ring}). See More on Algebra, Sections \ref{more-algebra-section-permanence-completion} and \ref{more-algebra-section-permanence-henselization} for the analogue of this material for the completion and henselization of a Noetherian local ring. \begin{lemma} \label{lemma-etale-dimension} Let $A$, $B$ be Noetherian local rings. Let $A \to B$ be a \'etale homomorphism of local rings. Then $\dim(A) = \dim(B)$. \end{lemma} \begin{proof} See for example Algebra, Lemma \ref{algebra-lemma-dimension-base-fibre-equals-total}. \end{proof} \begin{proposition} \label{proposition-etale-depth} Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be an \'etale homomorphism of local rings. Then $\text{depth}(A) = \text{depth}(B)$ \end{proposition} \begin{proof} See Algebra, Lemma \ref{algebra-lemma-apply-grothendieck}. \end{proof} \begin{proposition} \label{proposition-etale-CM} \begin{slogan} Being Cohen-Macaulay ascends and descends along \'etale maps. \end{slogan} Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be an \'etale homomorphism of local rings. Then $A$ is Cohen-Macaulay if and only if $B$ is so. \end{proposition} \begin{proof} A local ring $A$ is Cohen-Macaulay if and only $\dim(A) = \text{depth}(A)$. As both of these invariants is preserved under an \'etale extension, the claim follows. \end{proof} \begin{proposition} \label{proposition-etale-regular} Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be an \'etale homomorphism of local rings. Then $A$ is regular if and only if $B$ is so. \end{proposition} \begin{proof} If $B$ is regular, then $A$ is regular by Algebra, Lemma \ref{algebra-lemma-flat-under-regular}. Assume $A$ is regular. Let $\mathfrak m$ be the maximal ideal of $A$. Then $\dim_{\kappa(\mathfrak m)} \mathfrak m/\mathfrak m^2 = \dim(A) = \dim(B)$ (see Lemma \ref{lemma-etale-dimension}). On the other hand, $\mathfrak mB$ is the maximal ideal of $B$ and hence $\mathfrak m_B/\mathfrak m_B = \mathfrak mB/\mathfrak m^2B$ is generated by at most $\dim(B)$ elements. Thus $B$ is regular. (You can also use the slightly more general Algebra, Lemma \ref{algebra-lemma-flat-over-regular-with-regular-fibre}.) \end{proof} \begin{proposition} \label{proposition-etale-reduced} Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be an \'etale homomorphism of local rings. Then $A$ is reduced if and only if $B$ is so. \end{proposition} \begin{proof} It is clear from the faithful flatness of $A \to B$ that if $B$ is reduced, so is $A$. See also Algebra, Lemma \ref{algebra-lemma-descent-reduced}. Conversely, assume $A$ is reduced. By assumption $B$ is a localization of a finite type $A$-algebra $B'$ at some prime $\mathfrak q$. After replacing $B'$ by a localization we may assume that $B'$ is \'etale over $A$, see Lemma \ref{lemma-characterize-etale-Noetherian}. Then we see that Algebra, Lemma \ref{algebra-lemma-reduced-goes-up} applies to $A \to B'$ and $B'$ is reduced. Hence $B$ is reduced. \end{proof} \begin{remark} \label{remark-technicality-needed} The result on reducedness'' does not hold with a weaker definition of \'etale local ring maps $A \to B$ where one drops the assumption that $B$ is essentially of finite type over $A$. Namely, it can happen that a Noetherian local domain $A$ has nonreduced completion $A^\wedge$, see Examples, Section \ref{examples-section-local-completion-nonreduced}. But the ring map $A \to A^\wedge$ is flat, and $\mathfrak m_AA^\wedge$ is the maximal ideal of $A^\wedge$ and of course $A$ and $A^\wedge$ have the same residue fields. This is why it is important to consider this notion only for ring extensions which are essentially of finite type (or essentially of finite presentation if $A$ is not Noetherian). \end{remark} \begin{proposition} \label{proposition-etale-normal} Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be an \'etale homomorphism of local rings. Then $A$ is a normal domain if and only if $B$ is so. \end{proposition} \begin{proof} See Algebra, Lemma \ref{algebra-lemma-descent-normal} for descending normality. Conversely, assume $A$ is normal. By assumption $B$ is a localization of a finite type $A$-algebra $B'$ at some prime $\mathfrak q$. After replacing $B'$ by a localization we may assume that $B'$ is \'etale over $A$, see Lemma \ref{lemma-characterize-etale-Noetherian}. Then we see that Algebra, Lemma \ref{algebra-lemma-normal-goes-up} applies to $A \to B'$ and we conclude that $B'$ is normal. Hence $B$ is a normal domain. \end{proof} \noindent The preceeding propositions give some indication as to why we'd like to think of \'etale maps as local isomorphisms''. Another property that gives an excellent indication that we have the right'' definition is the fact that for $\mathbf{C}$-schemes of finite type, a morphism is \'etale if and only if the associated morphism on analytic spaces (the $\mathbf{C}$-valued points given the complex topology) is a local isomorphism in the analytic sense (open embedding locally on the source). This fact can be proven with the aid of the structure theorem and the fact that the analytification commutes with the formation of the completed local rings -- the details are left to the reader. \section{Descending \'etale morphisms} \label{section-descending-etale} \noindent In order to understand the language used in this section we encourage the reader to take a look at Descent, Section \ref{descent-section-descent-datum}. Let $f : X \to S$ be a morphism of schemes. Consider the pullback functor \begin{equation} \label{equation-descent-etale} \text{schemes }U\text{ \'etale over }S \longrightarrow \begin{matrix} \text{descent data }(V, \varphi)\text{ relative to }X/S \\ \text{ with }V\text{ \'etale over }X \end{matrix} \end{equation} sending $U$ to the canonical descent datum $(X \times_S U, can)$. \begin{lemma} \label{lemma-faithful} If $f : X \to S$ is surjective, then the functor (\ref{equation-descent-etale}) is faithful. \end{lemma} \begin{proof} Let $a, b : U_1 \to U_2$ be two morphisms between schemes \'etale over $S$. Assume the base changes of $a$ and $b$ to $X$ agree. We have to show that $a = b$. By Proposition \ref{proposition-equality} it suffices to show that $a$ and $b$ agree on points and residue fields. This is clear because for every $u \in U_1$ we can find a point $v \in X \times_S U_1$ mapping to $u$. \end{proof} \begin{lemma} \label{lemma-fully-faithful} Assume $f : X \to S$ is submersive and any \'etale base change of $f$ is submersive. Then the functor (\ref{equation-descent-etale}) is fully faithful. \end{lemma} \begin{proof} By Lemma \ref{lemma-faithful} the functor is faithful. Let $U_1 \to S$ and $U_2 \to S$ be \'etale morphisms and let $a : X \times_S U_1 \to X \times_S U_2$ be a morphism compatible with canonical descent data. We will prove that $a$ is the base change of a morphism $U_1 \to U_2$. \medskip\noindent Let $U'_2 \subset U_2$ be an open subscheme. Consider $W = a^{-1}(X \times_S U'_2)$. This is an open subscheme of $X \times_S U_1$ which is compatible with the canonical descent datum on $V_1 = X \times_S U_1$. This means that the two inverse images of $W$ by the projections $V_1 \times_{U_1} V_1 \to V_1$ agree. Since $V_1 \to U_1$ is surjective (as the base change of $X \to S$) we conclude that $W$ is the inverse image of some subset $U'_1 \subset U_1$. Since $W$ is open, our assumption on $f$ implies that $U'_1 \subset U_1$ is open. \medskip\noindent Let $U_2 = \bigcup U_{2, i}$ be an affine open covering. By the result of the preceding paragraph we obtain an open covering $U_1 = \bigcup U_{1, i}$ such that $X \times_S U_{1, i} = a^{-1}(X \times_S U_{2, i})$. If we can prove there exists a morphism $U_{1, i} \to U_{2, i}$ whose base change is the morphism $a_i : X \times_S U_{1, i} \to X \times_S U_{2, i}$ then we can glue these morphisms to a morphism $U_1 \to U_2$ (using faithfulness). In this way we reduce to the case that $U_2$ is affine. In particular $U_2 \to X$ is separated (Schemes, Lemma \ref{schemes-lemma-compose-after-separated}). \medskip\noindent Assume $U_2 \to S$ is separated. Then the graph $\Gamma_a$ of $a$ is a closed subscheme of $$V = (X \times_S U_1) \times_X (X \times_S U_2) = X \times_S U_1 \times_S U_2$$ by Schemes, Lemma \ref{schemes-lemma-semi-diagonal}. On the other hand the graph is open for example because it is a section of an \'etale morphism (Proposition \ref{proposition-properties-sections}). Since $a$ is a morphism of descent data, the two inverse images of $\Gamma_a \subset V$ under the projections $V \times_{U_1 \times_S U_2} V \to V$ are the same. Hence arguing as in the second paragraph of the proof we find an open and closed subscheme $\Gamma \subset U_1 \times_S U_2$ whose base change to $X$ gives $\Gamma_a$. Then $\Gamma \to U_1$ is an \'etale morphism whose base change to $X$ is an isomorphism. This means that $\Gamma \to U_1$ is universally bijective, hence an isomorphism by Theorem \ref{theorem-etale-radicial-open}. Thus $\Gamma$ is the graph of a morphism $U_1 \to U_2$ and the base change of this morphism is $a$ as desired. \end{proof} \begin{lemma} \label{lemma-fully-faithful-cases} Let $f : X \to S$ be a morphism of schemes. In the following cases the functor (\ref{equation-descent-etale}) is fully faithful: \begin{enumerate} \item $f$ is surjective and universally closed (e.g., finite, integral, or proper), \item $f$ is surjective and universally open (e.g., locally of finite presentation and flat, smooth, or etale), \item $f$ is surjective, quasi-compact, and flat. \end{enumerate} \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-fully-faithful}. For example a closed surjective map of topological spaces is submersive (Topology, Lemma \ref{topology-lemma-closed-morphism-quotient-topology}). Finite, integral, and proper morphisms are universally closed, see Morphisms, Lemmas \ref{morphisms-lemma-integral-universally-closed} and \ref{morphisms-lemma-finite-proper} and Definition \ref{morphisms-definition-proper}. On the other hand an open surjective map of topological spaces is submersive (Topology, Lemma \ref{topology-lemma-open-morphism-quotient-topology}). Flat locally finitely presented, smooth, and \'etale morphisms are universally open, see Morphisms, Lemmas \ref{morphisms-lemma-fppf-open}, \ref{morphisms-lemma-smooth-open}, and \ref{morphisms-lemma-etale-open}. The case of surjective, quasi-compact, flat morphisms follows from Morphisms, Lemma \ref{morphisms-lemma-fpqc-quotient-topology}. \end{proof} \begin{lemma} \label{lemma-reduce-to-affine} Let $f : X \to S$ be a morphism of schemes. Let $(V, \varphi)$ be a descent datum relative to $X/S$ with $V \to X$ \'etale. Let $S = \bigcup S_i$ be an open covering. Assume that \begin{enumerate} \item the pullback of the descent datum $(V, \varphi)$ to $X \times_S S_i/S_i$ is effective, \item the functor (\ref{equation-descent-etale}) for $X \times_S (S_i \cap S_j) \to (S_i \cap S_j)$ is fully faithful, and \item the functor (\ref{equation-descent-etale}) for $X \times_S (S_i \cap S_j \cap S_k) \to (S_i \cap S_j \cap S_k)$ is faithful. \end{enumerate} Then $(V, \varphi)$ is effective. \end{lemma} \begin{proof} (Recall that pullbacks of descent data are defined in Descent, Definition \ref{descent-definition-pullback-functor}.) Set $X_i = X \times_S S_i$. Denote $(V_i, \varphi_i)$ the pullback of $(V, \varphi)$ to $X_i/S_i$. By assumption (1) we can find an \'etale morphism $U_i \to S_i$ which comes with an isomorphism $X_i \times_{S_i} U_i \to V_i$ compatible with $can$ and $\varphi_i$. By assumption (2) we obtain isomorphisms $\psi_{ij} : U_i \times_{S_i} (S_i \cap S_j) \to U_j \times_{S_j} (S_i \cap S_j)$. By assumption (3) these isomorphisms satisfy the cocycle condition so that $(U_i, \psi_{ij})$ is a descend datum for the Zariski covering $\{S_i \to S\}$. Then Descent, Lemma \ref{descent-lemma-Zariski-refinement-coverings-equivalence} (which is essentially just a reformulation of Schemes, Section \ref{schemes-section-glueing-schemes}) tells us that there exists a morphism of schemes $U \to S$ and isomorphisms $U \times_S S_i \to U_i$ compatible with $\psi_{ij}$. The isomorphisms $U \times_S S_i \to U_i$ determine corresponding isomorphisms $X_i \times_S U \to V_i$ which glue to a morphism $X \times_S U \to V$ compatible with the canonical descent datum and $\varphi$. \end{proof} \begin{lemma} \label{lemma-split-henselian} Let $(A, I)$ be a henselian pair. Let $U \to \Spec(A)$ be a quasi-compact, separated, \'etale morphism such that $U \times_{\Spec(A)} \Spec(A/I) \to \Spec(A/I)$ is finite. Then $$U = U_{fin} \amalg U_{away}$$ where $U_{fin} \to \Spec(A)$ is finite and $U_{away}$ has no points lying over $Z$. \end{lemma} \begin{proof} By Zariski's main theorem, the scheme $U$ is quasi-affine. In fact, we can find an open immersion $U \to T$ with $T$ affine and $T \to \Spec(A)$ finite, see More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-finite-separated-pass-through-finite}. Write $Z = \Spec(A/I)$ and denote $U_Z \to T_Z$ the base change. Since $U_Z \to Z$ is finite, we see that $U_Z \to T_Z$ is closed as well as open. Hence by More on Algebra, Lemma \ref{more-algebra-lemma-characterize-henselian-pair} we obtain a unique decomposition $T = T' \amalg T''$ with $T'_Z = U_Z$. Set $U_{fin} = U \cap T'$ and $U_{away} = U \cap T''$. Since $T'_Z \subset U_Z$ we see that all closed points of $T'$ are in $U$ hence $T' \subset U$, hence $U_{fin} = T'$, hence $U_{fin} \to \Spec(A)$ is finite. We omit the proof of uniqueness of the decomposition. \end{proof} \begin{proposition} \label{proposition-effective} Let $f : X \to S$ be a surjective integral morphism. The functor (\ref{equation-descent-etale}) induces an equivalence $$\begin{matrix} \text{schemes quasi-compact,}\\ \text{separated, \'etale over }S \end{matrix} \longrightarrow \begin{matrix} \text{descent data }(V, \varphi)\text{ relative to }X/S\text{ with}\\ V\text{ quasi-compact, separated, \'etale over }X \end{matrix}$$ \end{proposition} \begin{proof} By Lemma \ref{lemma-fully-faithful-cases} the functor (\ref{equation-descent-etale}) is fully faithful and the same remains the case after any base change $S \to S'$. Let $(V, \varphi)$ be a descent data relative to $X/S$ with $V \to X$ quasi-compact, separated, and \'etale. We can use Lemma \ref{lemma-reduce-to-affine} to see that it suffices to prove the effectivity Zariski locally on $S$. In particular we may and do assume that $S$ is affine. \medskip\noindent If $S$ is affine we can find a directed set $\Lambda$ and an inverse system $X_\lambda \to S_\lambda$ of finite morphisms of affine schemes of finite type over $\Spec(\mathbf{Z})$ such that $(X \to S) = \lim (X_\lambda \to S_\lambda)$. See Algebra, Lemma \ref{algebra-lemma-limit-integral}. Since limits commute with limits we deduce that $X \times_S X = \lim X_\lambda \times_{S_\lambda} X_\lambda$ and $X \times_S X \times_S X = \lim X_\lambda \times_{S_\lambda} X_\lambda \times_{S_\lambda} X_\lambda$. Observe that $V \to X$ is a morphism of finite presentation. Using Limits, Lemmas \ref{limits-lemma-descend-finite-presentation} we can find an $\lambda$ and a descent datum $(V_\lambda, \varphi_\lambda)$ relative to $X_\lambda/S_\lambda$ whose pullback to $X/S$ is $(V, \varphi)$. Of course it is enough to show that $(V_\lambda, \varphi_\lambda)$ is effective. Note that $V_\lambda$ is quasi-compact by construction. After possibly increasing $\lambda$ we may assume that $V_\lambda \to X_\lambda$ is separated and \'etale, see Limits, Lemma \ref{limits-lemma-descend-separated-finite-presentation} and \ref{limits-lemma-descend-etale}. Thus we may assume that $f$ is finite surjective and $S$ affine of finite type over $\mathbf{Z}$. \medskip\noindent Consider an open $S' \subset S$ such that the pullback $(V', \varphi')$ of $(V, \varphi)$ to $X' = X \times_S S'$ is effective. Below we will prove, that $S' \not = S$ implies there is a strictly larger open over which the descent datum is effective. Since $S$ is Noetherian (and hence has a Noetherian underlying topological space) this will finish the proof. Let $\xi \in S$ be a generic point of an irreducible component of the closed subset $Z = S \setminus S'$. If $\xi \in S'' \subset S$ is an open over which the descent datum is effective, then the descent datum is effective over $S' \cup S''$ by the glueing argument of the first paragraph. Thus in the rest of the proof we may replace $S$ by an affine open neighbourhood of $\xi$. \medskip\noindent After a first such replacement we may assume that $Z$ is irreducible with generic point $Z$. Let us endow $Z$ with the reduced induced closed subscheme structure. After another shrinking we may assume $X_Z = X \times_S Z = f^{-1}(Z) \to Z$ is flat, see Morphisms, Proposition \ref{morphisms-proposition-generic-flatness}. Let $(V_Z, \varphi_Z)$ be the pullback of the descent datum to $X_Z/Z$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend} this descent datum is effective and we obtain an \'etale morphism $U_Z \to Z$ whose base change is isomorphic to $V_Z$ in a manner compatible with descent data. Of course $U_Z \to Z$ is quasi-compact and separated (Descent, Lemmas \ref{descent-lemma-descending-property-quasi-compact} and \ref{descent-lemma-descending-property-separated}). Thus after shrinking once more we may assume that $U_Z \to Z$ is finite, see Morphisms, Lemma \ref{morphisms-lemma-generically-finite}. \medskip\noindent Let $S = \Spec(A)$ and let $I \subset A$ be the prime ideal corresponding to $Z \subset S$. Let $(A^h, IA^h)$ be the henselization of the pair $(A, I)$. Denote $S^h = \Spec(A^h)$ and $Z^h = V(IA^h) \cong Z$. We claim that it suffices to show effectivity after base change to $S^h$. Namely, $\{S^h \to S, S' \to S\}$ is an fpqc covering ($A \to A^h$ is flat by More on Algebra, Lemma \ref{more-algebra-lemma-henselization-flat}) and by More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend} we have fpqc descent for separated \'etale morphisms. Namely, if $U^h \to S^h$ and $U' \to S'$ are the objects corresponding to the pullbacks $(V^h, \varphi^h)$ and $(V', \varphi')$, then the required isomorphisms $$U^h \times_S S^h \to S^h \times_S V^h \quad\text{and}\quad U^h \times_S S' \to S^h \times_S U'$$ are obtained by the fully faithfulness pointed out in the first paragraph. In this way we reduce to the situation described in the next paragraph. \medskip\noindent Here $S = \Spec(A)$, $Z = V(I)$, $S' = S \setminus Z$ where $(A, I)$ is a henselian pair, we have $U' \to S'$ corresponding to the descent datum $(V', \varphi')$ and we have a finite \'etale morphism $U_Z \to Z$ corresponding to the descent datum $(V_Z, \varphi_Z)$. We no longer have that $A$ is of finite type over $\mathbf{Z}$; but the rest of the argument will not even use that $A$ is Noetherian. By More on Algebra, Lemma \ref{more-algebra-lemma-finite-etale-equivalence} we can find a finite \'etale morphism $U_{fin} \to S$ whose restriction to $Z$ is isomorphic to $U_Z \to Z$. Write $X = \Spec(B)$ and $Y = V(IB)$. Since $(B, IB)$ is a henselian pair (More on Algebra, Lemma \ref{more-algebra-lemma-integral-over-henselian-pair}) and since the restriction $V \to X$ to $Y$ is finite (as base change of $U_Z \to Z$) we see that there is a canonical disjoint union decomposition $$V = V_{fin} \amalg V_{away}$$ were $V_{fin} \to X$ is finite and where $V_{away}$ has no points lying over $Y$. See Lemma \ref{lemma-split-henselian}. Using the uniqueness of this decomposition over $X \times_S X$ we see that $\varphi$ preserves it and we obtain $$(V, \varphi) = (V_{fin}, \varphi_{fin}) \amalg (V_{away}, \varphi_{away})$$ in the category of descent data. By More on Algebra, Lemma \ref{more-algebra-lemma-finite-etale-equivalence} there is a unique isomorphism $$X \times_S U_{fin} \longrightarrow V_{fin}$$ compatible with the given isomorphism $Y \times_Z U_Z \to V \times_X Y$ over $Y$. By the uniqueness we see that this isomorphism is compatible with descent data, i.e., $(X \times_S U_{fin}, can) \cong (V_{fin}, \varphi_{fin})$. Denote $U'_{fin} = U_{fin} \times_S S'$. By fully faithfulness we obtain a morphism $U'_{fin} \to U'$ which is the inclusion of an open (and closed) subscheme. Then we set $U = U_{fin} \amalg_{U'_{fin}} U'$ (glueing of schemes as in Schemes, Section \ref{schemes-section-glueing-schemes}). The morphisms $X \times_S U_{fin} \to V$ and $X \times_S U' \to V$ glue to a morphism $X \times_S U \to V$ which is the desired isomorphism. \end{proof} \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 composition $$Z_n \to D^\nu \times_D \ldots \times_D D^\nu \to \ldots \to D^\nu \times_D D^\nu \to D^\nu \to D \to X$$ 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). Similarly, $Z_1 = D^\nu \to X$ 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 verifications. \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} \input{chapters} \bibliography{my} \bibliographystyle{amsalpha} \end{document}