Introduction
In this Chapter, we discuss étale 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 étale cohomology. An auxiliary goal is to provide enough evidence to ensure that the reader stops calling the phrase “the étale topology of schemes” an exercise in general nonsense, if (s)he does indulge in such blasphemy.
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.
Conventions
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:
A ring homomorphism \(A \to B\) of finite type with \(A\) Noetherian is of finite presentation. See Algebra, Lemma 00FP.
A morphism (locally) of finite type between locally Noetherian schemes is automatically (locally) of finite presentation. See Morphisms, Lemma 01TX.
Add more like this here.
Unramified morphisms
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 00US) and it is different. After discussing the notion a bit we globalize it to describe unramified morphisms of locally Noetherian schemes.
Definition
Let \(A\), \(B\) be Noetherian local rings. A local homomorphism \(A \to B\) is said to be unramified homomorphism of local rings if
\(\mathfrak m_AB = \mathfrak m_B\),
\(\kappa(\mathfrak m_B)\) is a finite separable extension of \(\kappa(\mathfrak m_A)\), and
\(B\) is essentially of finite type over \(A\) (this means that \(B\) is the localization of a finite type \(A\)-algebra at a prime).
This is the local version of the definition in Algebra, Section 00US. 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 00UW and 02FM 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.
Lemma
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.
Proof
See discussion above.
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 0316.
Lemma
Let \(A\), \(B\) be Noetherian local rings. Let \(A \to B\) be a local homomorphism.
if \(A \to B\) is an unramified homomorphism of local rings, then \(B^\wedge\) is a finite \(A^\wedge\) module,
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,
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,
if \(A\) and \(B\) are complete discrete valuation rings, then \(A \to B\) is an unramified homomorphism of local rings if and only if 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 \(A\)).
Proof
Part (1) is a special case of Algebra, Lemma 0394. 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 00DV) 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.
Lemma
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
\(A \to B\) is an unramified homomorphism of local rings
\(A^\wedge \to B^\wedge\) is an unramified homomorphism of local rings, and
\(A^\wedge \to B^\wedge\) is unramified.
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 \(\mathfrak m_AB = \mathfrak m_B\).
Assume the equivalent conditions (1) and (2). By Lemma 039H we see that \(A^\wedge \to B^\wedge\) is finite. Hence \(A^\wedge \to B^\wedge\) is of finite presentation, and by Algebra, Lemma 02FM 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.
Assume (3). By Algebra, Lemma 00UW we conclude that \(A^\wedge \to B^\wedge\) is an unramified homomorphism of local rings, i.e., (2) holds.
Definition
(See Morphisms, Definition 02G4 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 \(x \in X\).
We say \(f\) is unramified at \(x\) if \(\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}\) is an unramified homomorphism of local rings.
The morphism \(f : X \to Y\) is said to be unramified if it is unramified at all points of \(X\).
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.
Lemma
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 024N if and only if it is unramified in the sense of Morphisms, Definition 02G4.
Proof
This follows from Lemma 039G and the definitions.
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:
Unramifiedness is local on the source and the target in the Zariski topology.
Unramified morphisms are stable under base change and composition. See Morphisms, Lemmas 02GA and 02G9.
Unramified morphisms of schemes are locally quasi-finite and quasi-compact unramified morphisms are quasi-finite. See Morphisms, Lemma 02V5
Unramified morphisms have relative dimension \(0\). See Morphisms, Definition 02NJ and Morphisms, Lemma 0397.
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 02G8.
Let \(X\) and \(Y\) be unramified over a base scheme \(S\). Any \(S\)-morphism from \(X\) to \(Y\) is unramified. See Morphisms, Lemma 02GG.
Three other characterizations of unramified morphisms
The following theorem gives three equivalent notions of being unramified at a point. See Morphisms, Lemma 02GF for (part of) the statement for general schemes.
Theorem
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
\(f\) is unramified at \(x\),
the stalk \(\Omega_{X/Y, x}\) of the module of relative differentials at \(x\) is trivial,
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
the diagonal \(\Delta_{X/Y} : X \to X \times_Y X\) is a local isomorphism at \(x\).
Proof
The equivalence of (1) and (2) is proved in Morphisms, Lemma 02GF.
If \(f\) is unramified at \(x\), then \(f\) is unramified in an open neighbourhood of \(x\); this does not follow immediately from Definition 024N of this chapter but it does follow from Morphisms, Definition 02G4 which we proved to be equivalent in Lemma 039J. 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 02GE the morphism \(U \to U \times_V U\) is an open immersion. This proves that (1) implies (4).
If \(\Delta_{X/Y}\) is a local isomorphism at \(x\), then \(\Omega_{X/Y, x} = 0\) by Morphisms, Lemma 08S2. Hence we see that (4) implies (2). At this point we know that (1), (2) and (4) are all equivalent.
Assume (3). The assumption on the diagram combined with Morphisms, Lemma 01UZ show that \(\Omega_{U/V, x} = 0\). Since \(\Omega_{U/V, x} = \Omega_{X/Y, x}\) we conclude (2) holds.
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 01UT 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 01OW and Morphisms, Lemma 01T2. 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 00RT). 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.
How can we use this theorem? Well, here are a few remarks:
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 01UX. The theorem therefore implies that if \(g \circ f\) is unramified, then so is \(f\). This is Morphisms, Lemma 02GG.
Since \(\Omega_{X/Y}\) is isomorphic to the conormal sheaf of the diagonal morphism (Morphisms, Lemma 08S2) 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 02GB and 02GC.
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”.
The functorial characterization of unramified morphisms
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.
Theorem
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:
\(f\) is unramified,
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.
Proof
See More on Morphisms, Lemma 02HE for a more general statement and proof. What follows is a sketch of the proof in the current case.
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 024P.
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.
Topological properties of unramified morphisms
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.
Proposition
Sections of unramified morphisms.
Any section of an unramified morphism is an open immersion.
Any section of a separated morphism is a closed immersion.
Any section of an unramified separated morphism is open and closed.
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 01KK (resp. Morphisms, Lemma 02GE). Hence so is the graph as a base change (by Schemes, Lemma 01JY). In the special case \(X' = S\), we obtain (1), resp. (2). Part (3) follows on combining (1) and (2).
We can now explicitly describe the sections of unramified morphisms.
Theorem
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 \(x\).
Proof
Direct from Proposition 024T part (3).
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 [SGA1, Exposé V]). See also the more general Morphisms, Lemma 04HB.
Proposition
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
\(Y\) is connected
\(x = f(y) = g(y)\), and
the induced maps \(f^\sharp, g^\sharp : \kappa(x) \to \kappa(y)\) on residue fields are equal.
Then \(f = g\).
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 02GA and Schemes, Lemma 01KU). Thus according to Theorem 024U 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 \(f'(y) = g'(y) \in X_Y\).
Lemma
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.
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 \(\kappa(x_i)/\kappa(s)\) is a finite field extension. See Morphisms, Lemma 02V5, 01TG, and 02NH. 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 024V.
General case. There exists a nonempty open \(U \subset S\) such that \(X_U \to U\) is finite (in particular separated), see Morphisms, Lemma 02NW (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 01T7). 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 02GE.
Universally injective, unramified morphisms
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 01S3. Universally injective and unramified morphisms can be characterized as follows.
Lemma
Let \(f : X \to S\) be a morphism of schemes. The following are equivalent:
\(f\) is unramified and a monomorphism,
\(f\) is unramified and universally injective,
\(f\) is locally of finite type and a monomorphism,
\(f\) is universally injective, locally of finite type, and formally unramified,
\(f\) is locally of finite type and \(X_s\) is either empty or \(X_s \to s\) is an isomorphism for all \(s \in S\).
Proof
We have seen in More on Morphisms, Lemma 02HE 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 02GE) and surjective (Morphisms, Lemma 01S4) hence an isomorphism, i.e., \(f\) is a monomorphism. In this way we see that (2) implies (1).
Condition (3) implies (5) because monomorphisms are preserved under base change (Schemes, Lemma 02YC) and because of the description of monomorphisms towards the spectra of fields in Schemes, Lemma 03DP. Condition (5) implies (4) by Morphisms, Lemmas 01S4 and 02G8.
This leads to the following useful characterization of closed immersions.
Lemma
Let \(f : X \to S\) be a morphism of schemes. The following are equivalent:
\(f\) is a closed immersion,
\(f\) is a proper monomorphism,
\(f\) is proper, unramified, and universally injective,
\(f\) is universally closed, unramified, and a monomorphism,
\(f\) is universally closed, unramified, and universally injective,
\(f\) is universally closed, locally of finite type, and a monomorphism,
\(f\) is universally closed, universally injective, locally of finite type, and formally unramified.
Proof
The equivalence of (4) – (7) follows immediately from Lemma 05VH.
Let \(f : X \to S\) satisfy (6). Then \(f\) is separated, see Schemes, Lemma 01L4 and has finite fibres. Hence More on Morphisms, Lemma 02LS shows \(f\) is finite. Then Morphisms, Lemma 03BB implies \(f\) is a closed immersion, i.e., (1) holds.
Note that (1) \(\Rightarrow\) (2) because a closed immersion is proper and a monomorphism (Morphisms, Lemma 01W5 and Schemes, Lemma 01L7). By Lemma 05VH we see that (2) implies (3). It is clear that (3) implies (5).
Here is another result of a similar flavor.
Lemma
Let \(\pi : X \to S\) be a morphism of schemes. Let \(s \in S\). Assume that
\(\pi\) is finite,
\(\pi\) is unramified,
\(\pi^{-1}(\{s\}) = \{x\}\), and
\(\kappa(s) \subset \kappa(x)\) is purely inseparable1.
Then there exists an open neighbourhood \(U\) of \(s\) such that \(\pi|_{\pi^{-1}(U)} : \pi^{-1}(U) \to U\) is a closed immersion.
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 \(x\). Conditions (2), (3) and (4) imply that \(B_{\mathfrak q}/\mathfrak pB_{\mathfrak q} = \kappa(\mathfrak p)\). By Algebra, Lemma 00EA we have \(B_{\mathfrak q} = B_{\mathfrak p}\) (note that a finite ring map satisfies going up, see Algebra, Section 00HU.) 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 00DV) 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.
The topological results presented above will be used to give a functorial characterization of étale morphisms similar to Theorem 024R.
Examples of unramified morphisms
Here are a few examples.
Example
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 étale if and only if it is unramified. Thus, in this case at least, we obtain a very easy description of the étale topology of a scheme. Of course, the cohomology of this topology is another story.
Example
Property (3) in Theorem 024P 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 étale map (see Algebra, Definition 00T6, Lemma 00T7 and Example 00T8).
Example
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 \(X \to \Spec(\mathcal{O}_K)\) 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 étale. 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 étale morphism \(X \to Y\). Thus, this is an example of a finite étale covering.
Flat morphisms
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.
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.
Definition
Flatness of modules and rings.
A module \(N\) over a ring \(A\) is said to be flat if the functor \(M \mapsto M \otimes_A N\) is exact.
If this functor is also faithful, we say that \(N\) is faithfully flat over \(A\).
A morphism of rings \(f : A \to B\) is said to be flat (resp. faithfully flat) if the functor \(M \mapsto M \otimes_A B\) is exact (resp. faithful and exact).
Here is a list of facts with references to the algebra chapter.
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.
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 00HT.
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 00HI.
Finite flat modules over local rings are free. See Algebra, Lemma 00NZ.
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 00HT
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 00HR.
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 00HQ.
If \(A\) is a Noetherian local ring, the completion \(A^\wedge\) is faithfully flat over \(A\). See Algebra, Lemma 00MC.
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 00HJ.)
Before we move on to the geometric category, we present Grothendieck’s theorem, which provides a convenient recipe for producing flat modules.
Theorem
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.
Proof
See Algebra, Lemma 00ME. See also [MatCA, Section 20].
Definition
(See Morphisms, Definition 01U3). Let \(f : X \to Y\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module.
Let \(x \in X\). We say \(\mathcal{F}\) is 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.
Let \(x \in X\). We say \(f\) is flat at \(x \in X\) if \(\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}\) is flat.
We say \(f\) is flat if it is flat at all points of \(X\).
A morphism \(f : X \to Y\) that is flat and surjective is sometimes said to be faithfully flat.
Once again, here is a list of results:
The property (of a morphism) of being flat is, by fiat, local in the Zariski topology on the source and the target.
Open immersions are flat. (This is clear because it induces isomorphisms on local rings.)
Flat morphisms are stable under base change and composition. Morphisms, Lemmas 01U9 and 01U7.
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.
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 01PE to see that \(Y\) has “enough” ideal sheaves and use the characterization of flatness in Algebra, Lemma 00HD.)
Topological properties of flat morphisms
We “recall” below some openness properties that flat morphisms enjoy.
Theorem
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 \(X\).
Proof
See More on Morphisms, Theorem 0399.
Theorem
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.
Proof
See Morphisms, Lemma 01UA.
Theorem
A faithfully flat quasi-compact morphism is a quotient map for the Zariski topology.
Proof
See Morphisms, Lemma 02JY.
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.
Étale morphisms
In this section, we will define étale morphisms and prove a number of important properties about them. The most important one, no doubt, is the functorial characterization presented in Theorem 025K. Following this, we will also discuss a few properties of rings which are insensitive to an étale extension (properties which hold for a ring if and only if they hold for all its étale extensions) to motivate the basic tenet of étale cohomology – étale morphisms are the algebraic analogue of local isomorphisms.
As the title suggests, we will define the class of étale 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 étale site \(S_\etale\). Intuitively, an étale 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 étale (in the aforementioned sense). But, rather than proceeding via such aesthetically displeasing constructions, we will adopt a cleaner, albeit slightly more abstract, algebraic approach.
We first define “étale homomorphisms of local rings” for Noetherian local rings. We cannot use the term “étale”, as there already is a notion of an étale ring map (Algebra, Section 00U0) and it is different.
Definition
Let \(A\), \(B\) be Noetherian local rings. A local homomorphism \(f : A \to B\) is said to be an étale homomorphism of local rings if it is flat and an unramified homomorphism of local rings (please see Definition 024M).
This is the local version of the definition of an étale ring map in Algebra, Section 00U0. 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 00U9) that an étale \(R\)-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.
Lemma
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 étale at \(\mathfrak q\) if and only if \(A_{\mathfrak p} \to B_{\mathfrak q}\) is an étale homomorphism of local rings.
Proof
See Algebra, Lemmas 00U2 (flatness of étale maps), 00U4 (étale maps are unramified) and 00U6 (flat and unramified maps are étale).
Lemma
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
\(A \to B\) is an étale homomorphism of local rings
\(A^\wedge \to B^\wedge\) is an étale homomorphism of local rings, and
\(A^\wedge \to B^\wedge\) is étale.
Moreover, in this case \(B^\wedge \cong (A^\wedge)^{\oplus n}\) as \(A^\wedge\)-modules for some \(n \geq 1\).
Proof
To see the equivalences of (1), (2) and (3), as we have the corresponding results for unramified ring maps (Lemma 039I) 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 039H 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 0250.
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.
Definition
(See Morphisms, Definition 02GI.) Let \(Y\) be a locally Noetherian scheme. Let \(f : X \to Y\) be a morphism of schemes which is locally of finite type.
Let \(x \in X\). We say \(f\) is étale at \(x \in X\) if \(\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}\) is an étale homomorphism of local rings.
The morphism is said to be étale if it is étale at all its points.
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 étale is open.
Lemma
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 étale at \(x\) in the sense of Definition 0259 if and only if it is étale at \(x\) in the sense of Morphisms, Definition 02GI.
Proof
This follows from Lemma 039L and the definitions.
Here are some results on étale 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:
An étale morphism is unramified. (Clear from our definitions.)
Étaleness is local on the source and the target in the Zariski topology.
Étale morphisms are stable under base change and composition. See Morphisms, Lemmas 02GO and 02GN.
Étale morphisms of schemes are locally quasi-finite and quasi-compact étale morphisms are quasi-finite. (This is true because it holds for unramified morphisms as seen earlier.)
Étale morphisms have relative dimension \(0\). See Morphisms, Definition 02NJ and Morphisms, Lemma 0397.
A morphism is étale if and only if it is flat and all its fibres are étale. See Morphisms, Lemma 02GM.
Étale morphisms are open. This is true because an étale morphism is flat, and Theorem 039K.
Let \(X\) and \(Y\) be étale over a base scheme \(S\). Any \(S\)-morphism from \(X\) to \(Y\) is étale. See Morphisms, Lemma 02GW.
The structure theorem
We present a theorem which describes the local structure of étale and unramified morphisms. Besides its obvious independent importance, this theorem also allows us to make the transition to another definition of étale morphisms that captures the geometric intuition better than the one we’ve used so far.
To state it we need the notion of a standard étale ring map, see Algebra, Definition 00UB. Namely, suppose that \(R\) is a ring and \(f, g \in R[t]\) are polynomials such that
\(f\) is a monic polynomial, and
\(f' = \text{d}f/\text{d}t\) is invertible in the localization \(R[t]_g/(f)\).
Then the map \[R \longrightarrow R[t]_g/(f) = R[t, 1/g]/(f)\] is a standard étale algebra, and any standard étale 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 étale (as expected of course). A special case of a standard étale 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 étale algebra is (isomorphic to) a principal localization of one of these.
Theorem
Let \(f : A \to B\) be an étale homomorphism of local rings. Then there exist \(f, g \in A[t]\) such that
\(B' = A[t]_g/(f)\) is standard étale – see (a) and (b) above, and
\(B\) is isomorphic to a localization of \(B'\) at a prime.
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 039L we see that \(A \to B'\) is étale at \(\mathfrak q\). Hence we may apply Algebra, Proposition 00UE to see that a principal localization of \(B'\) is standard étale.
Here is the version for unramified homomorphisms of local rings.
Theorem
Let \(f : A \to B\) be an unramified morphism of local rings. Then there exist \(f, g \in A[t]\) such that
\(B' = A[t]_g/(f)\) is standard étale – see (a) and (b) above, and
\(B\) is isomorphic to a quotient of a localization of \(B'\) at a prime.
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 039G we see that \(A \to B'\) is unramified at \(\mathfrak q\). Hence we may apply Algebra, Proposition 0395 to see that a principal localization of \(B'\) is a quotient of a standard étale \(A\)-algebra.
Via standard lifting arguments, one then obtains the following geometric statement which will be of essential use to us.
Theorem
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 étale 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.
Proof
This is equivalent to Morphisms, Lemma 02GT although the statements differ slightly. See also, Varieties, Lemma 0CBJ for a variant for unramified morphisms.
Étale and smooth morphisms
An étale 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 étale locally of this form. Here is the precise statement.
Theorem
Let \(\varphi : X \to Y\) be a morphism of schemes. Let \(x \in X\). If \(\varphi\) is smooth at \(x\), then there exist an 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 étale.
Proof
See Morphisms, Lemma 054L.
Topological properties of étale morphisms
We present a few of the topological properties of étale and unramified morphisms. First, we give what Grothendieck calls the fundamental property of étale morphisms, see [SGA1, Exposé I.5].
Theorem
Let \(f : X \to Y\) be a morphism of schemes. The following are equivalent:
\(f\) is an open immersion,
\(f\) is universally injective and étale, and
\(f\) is a flat monomorphism, locally of finite presentation.
Proof
An open immersion is universally injective since any base change of an open immersion is an open immersion. Moreover, it is étale by Morphisms, Lemma 02GP. Hence (1) implies (2).
Assume \(f\) is universally injective and étale. Since \(f\) is étale it is flat and locally of finite presentation, see Morphisms, Lemmas 02GS and 02GR. By Lemma 05VH we see that \(f\) is a monomorphism. Hence (2) implies (3).
Assume \(f\) is flat, locally of finite presentation, and a monomorphism. Then \(f\) is open, see Morphisms, Lemma 01UA. 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 06NC shows that \(f\) is an isomorphism and we win.
Lemma
Let \(f : T \to S\) be a surjective monomorphism of finite type, where \(S\) is locally Noetherian.
The morphism \(f\) is an isomorphism if and only if, for every \(s \in S\) and every \(n \geq 0\), its base change to \[S_{s, n} = \Spec(\mathcal{O}_{S, s}/\mathfrak m_s^{n + 1})\] is an isomorphism.
If \(S\) is reduced, then \(f\) is an isomorphism if and only if, for every morphism \(\Spec(V) \to S\) from the spectrum of a discrete valuation ring, the base change \(T_V \to \Spec(V)\) has a section.
Proof
Only the reverse implications require proof. Suppose first that the infinitesimal neighbourhood condition in (1) holds. For \(s \in S\), let \(t \in T\) be the unique point over \(s\) and put \[A = \mathcal{O}_{S, s}, \qquad B = \mathcal{O}_{T, t}.\] The case \(n = 0\) identifies the residue fields. For every \(n\), the asserted isomorphism after base change gives \[A/\mathfrak m_A^{n + 1} \longrightarrow B/\mathfrak m_A^{n + 1}B\] as an isomorphism. In particular \(\mathfrak m_B = \mathfrak m_A B\), and passing to inverse limits shows that \(A^\wedge \to B^\wedge\) is an isomorphism. The local rings are Noetherian and \(B\) is essentially of finite type over \(A\). Thus Lemma 039M shows that \(A \to B\) is étale. Hence \(f\) is étale. An étale monomorphism is an open immersion by Theorem 025G; since \(f\) is surjective, it is an isomorphism.
Assume now that \(S\) is reduced and the condition in (2) holds. Consider a valuative diagram for \(f\) with valuation ring a discrete valuation ring \(V\). The section of \(T_V \to \Spec(V)\) restricts on the generic point to the given map: both maps have the same composition with the monomorphism \(T_V \to \Spec(V)\), so they agree. It is also the unique lift. The Noetherian discrete valuation criterion, Limits, Lemma 0208, therefore shows that \(f\) is proper. A proper monomorphism is a closed immersion by Lemma 04XV. Its image is all of \(S\). The defining ideal is consequently locally nilpotent, and it is zero because \(S\) is reduced. Thus \(f\) is an isomorphism.
Here is another result of a similar flavor.
Lemma
Let \(\pi : X \to S\) be a morphism of schemes. Let \(s \in S\). Assume that
\(\pi\) is finite,
\(\pi\) is étale,
\(\pi^{-1}(\{s\}) = \{x\}\), and
\(\kappa(s) \subset \kappa(x)\) is purely inseparable2.
Then there exists an open neighbourhood \(U\) of \(s\) such that \(\pi|_{\pi^{-1}(U)} : \pi^{-1}(U) \to U\) is an isomorphism.
Proof
By Lemma 04DG 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 étale and a closed immersion is an open immersion (for example by Theorem 025G). Hence after shrinking \(U\) we obtain an isomorphism.
Lemma
Let \(U \to X\) be an étale morphism of schemes where \(X\) is a scheme in characteristic \(p\). Then the relative Frobenius \(F_{U/X} : U \to U \times_{X, F_X} X\) is an isomorphism.
Proof
The morphism \(F_{U/X}\) is a universal homeomorphism by Varieties, Lemma 0CCB. The morphism \(F_{U/X}\) is étale as a morphism between schemes étale over \(X\) (Morphisms, Lemma 02GW). Hence \(F_{U/X}\) is an isomorphism by Theorem 025G.
Topological invariance of the étale topology
Next, we present an extremely crucial theorem which, roughly speaking, says that étaleness is a topological property.
Theorem
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 étale over \(S\), then the map \[\Mor_S(Y, X) \longrightarrow \Mor_{S_0}(Y_0, X_0)\] is bijective.
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 étale structure morphism \(f : X \to S\).
Uniqueness. Suppose we have two sections \(\sigma, \sigma'\) through which \(\sigma_0\) factors. Because \(X \to S\) is étale we see that \(\Delta : X \to X \times_S X\) is an open immersion (Morphisms, Lemma 02GE). 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 \(\sigma = \sigma'\).
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.
Existence when \(S\) and \(X\) are affine. Write \(S = \Spec(A)\) and \(X = \Spec(B)\). Then \(A \to B\) is étale 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 07K4.
From the proof of preceding theorem, we also obtain one direction of the promised functorial characterization of étale morphisms. The following theorem will be strengthened in Étale Cohomology, Theorem 04DZ.
Theorem
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 \[\{ \text{schemes }X\text{ \'etale over }S \} \leftrightarrow \{ \text{schemes }X_0\text{ \'etale over }S_0 \}\]
Proof
By Theorem 025H 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 étale morphism of schemes.
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 étale 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 01JA. Applying Schemes, Lemma 01JC 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 étale and \(X_0 \cong Y\) by construction.
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 00U9 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 0544). 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 0AMG. Hence \(R \to A\) is étale and the proof is complete.
The functorial characterization
We finally present the promised functorial characterization. Thus there are four ways to think about étale morphisms of schemes:
as a smooth morphism of relative dimension \(0\),
as locally finitely presented, flat, and unramified morphisms,
using the structure theorem, and
using the functorial characterization.
Theorem
Let \(f : X \to S\) be a morphism that is locally of finite presentation. The following are equivalent
\(f\) is étale,
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.
Proof
This is More on Morphisms, Lemma 02HM.
This characterization says that solutions to the equations defining \(X\) can be lifted uniquely through nilpotent thickenings.
Étale local structure of unramified morphisms
In the chapter More on Morphisms, Section 04HF the reader can find some results on the étale 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 (02LJ). We start with a very general case.
Lemma
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 étale 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
\(V_{i, j} \to U\) is a closed immersion passing through \(u\),
\(u\) is not in the image of \(V_{i, j} \cap V_{i', j'}\) unless \(i = i'\) and \(j = j'\), and
any point of \((X_U)_u\) mapping to \(x_i\) is in some \(V_{i, j}\).
Proof
By Morphisms, Definition 02G4 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 02LM to get the étale neighbourhood \((U, u)\) and the opens \(V_{i, j}\) finite over \(U\). By Lemma 04DG after possibly shrinking \(U\) we get that \(V_{i, j} \to U\) is a closed immersion.
Lemma
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 étale neighbourhood \((U, u) \to (S, s)\) and a disjoint union decomposition \[X_U = W \amalg \coprod\nolimits_{i, j} V_{i, j}\] such that
\(V_{i, j} \to U\) is a closed immersion passing through \(u\),
the fibre \(W_u\) contains no point mapping to any \(x_i\).
In particular, if \(f^{-1}(\{s\}) = \{x_1, \ldots, x_n\}\), then the fibre \(W_u\) is empty.
Proof
Apply Lemma 04HH. 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 01W6. 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.
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 étale locally on the base a closed immersion.
Lemma
Let \(f : X \to S\) be a finite unramified morphism of schemes. Let \(s \in S\). There exists an étale neighbourhood \((U, u) \to (S, s)\) and a finite disjoint union decomposition \[X_U = \coprod\nolimits_j V_j\] such that each \(V_j \to U\) is a closed immersion.
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 04HI to this set (a finite morphism is separated, see Morphisms, Section 01WG). 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.
Étale local structure of étale morphisms
This is a bit silly, but perhaps helps form intuition about étale morphisms. We simply copy over the results of Section 04HG and change “closed immersion” into “isomorphism”.
Lemma
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 étale at each \(x_i\). Then there exists an étale 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
\(V_{i, j} \to U\) is an isomorphism,
\(u\) is not in the image of \(V_{i, j} \cap V_{i', j'}\) unless \(i = i'\) and \(j = j'\), and
any point of \((X_U)_u\) mapping to \(x_i\) is in some \(V_{i, j}\).
Proof
An étale morphism is unramified, hence we may apply Lemma 04HH. Now \(V_{i, j} \to U\) is a closed immersion and étale. Hence it is an open immersion, for example by Theorem 025G. Replace \(U\) by the intersection of the images of \(V_{i, j} \to U\) to get the lemma.
Lemma
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 étale at each \(x_i\). Then there exists an étale neighbourhood \((U, u) \to (S, s)\) and a finite disjoint union decomposition \[X_U = W \amalg \coprod\nolimits_{i, j} V_{i, j}\] of schemes such that
\(V_{i, j} \to U\) is an isomorphism,
the fibre \(W_u\) contains no point mapping to any \(x_i\).
In particular, if \(f^{-1}(\{s\}) = \{x_1, \ldots, x_n\}\), then the fibre \(W_u\) is empty.
Proof
An étale morphism is unramified, hence we may apply Lemma 04HI. As in the proof of Lemma 04HL the morphisms \(V_{i, j} \to U\) are open immersions and we win after replacing \(U\) by the intersection of their images.
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 étale morphism is étale locally on the base a “topological covering space”, i.e., a finite product of copies of the base.
Lemma
Let \(f : X \to S\) be a finite étale morphism of schemes. Let \(s \in S\). There exists an étale neighbourhood \((U, u) \to (S, s)\) and a finite disjoint union decomposition \[X_U = \coprod\nolimits_j V_j\] of schemes such that each \(V_j \to U\) is an isomorphism.
Proof
An étale morphism is unramified, hence we may apply Lemma 04HJ. As in the proof of Lemma 04HL we see that \(V_{i, j} \to U\) is an open immersion and we win after replacing \(U\) by the intersection of their images.
Permanence properties
In what follows, we present a few “permanence” properties of étale homomorphisms of Noetherian local rings (as defined in Definition 0258). See More on Algebra, Sections 07NU and 07QL for the analogue of this material for the completion and henselization of a Noetherian local ring.
Lemma
Let \(A\), \(B\) be Noetherian local rings. Let \(A \to B\) be a étale homomorphism of local rings. Then \(\dim(A) = \dim(B)\).
Proof
See for example Algebra, Lemma 00ON.
Proposition
Let \(A\), \(B\) be Noetherian local rings. Let \(f : A \to B\) be an étale homomorphism of local rings. Then \(\text{depth}(A) = \text{depth}(B)\)
Proof
See Algebra, Lemma 0337.
Proposition
Let \(A\), \(B\) be Noetherian local rings. Let \(f : A \to B\) be an étale homomorphism of local rings. Then \(A\) is Cohen-Macaulay if and only if \(B\) is so.
Proof
A local ring \(A\) is Cohen-Macaulay if and only if \(\dim(A) = \text{depth}(A)\). As both of these invariants is preserved under an étale extension, the claim follows.
Proposition
Let \(A\), \(B\) be Noetherian local rings. Let \(f : A \to B\) be an étale homomorphism of local rings. Then \(A\) is regular if and only if \(B\) is so.
Proof
If \(B\) is regular, then \(A\) is regular by Algebra, Lemma 00OF. 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 039S). 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 031E.)
Proposition
Let \(A\), \(B\) be Noetherian local rings. Let \(f : A \to B\) be an étale homomorphism of local rings. Then \(A\) is reduced if and only if \(B\) is so.
Proof
It is clear from the faithful flatness of \(A \to B\) that if \(B\) is reduced, so is \(A\). See also Algebra, Lemma 033F. 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 étale over \(A\), see Lemma 039L. Then we see that Algebra, Lemma 033B applies to \(A \to B'\) and \(B'\) is reduced. Hence \(B\) is reduced.
Remark
The result on “reducedness” does not hold with a weaker definition of étale 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 02JD. 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).
Proposition
Let \(A\), \(B\) be Noetherian local rings. Let \(f : A \to B\) be an étale homomorphism of local rings. Then \(A\) is a normal domain if and only if \(B\) is so.
Proof
See Algebra, Lemma 033G 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 étale over \(A\), see Lemma 039L. Then we see that Algebra, Lemma 033C applies to \(A \to B'\) and we conclude that \(B'\) is normal. Hence \(B\) is a normal domain.
The preceding propositions give some indication as to why we’d like to think of étale maps as “local isomorphisms”. The following comparison gives another indication that we have the right definition.
Lemma
Let \(f : X \to Y\) be a morphism of schemes of finite type over \(\mathbf{C}\). Then \(f\) is étale if and only if the associated morphism \[f^{an} : X^{an} \longrightarrow Y^{an}\] of complex analytic spaces is a local isomorphism.
Proof
This is the local analytic criterion for étaleness; see [SGA1, Exposé XII]. It can also be proved from the structure theorem for étale morphisms and the compatibility of analytification with completed local rings.
Descending étale morphisms
In order to understand the language used in this section we encourage the reader to take a look at Descent, Section 023U. Let \(f : X \to S\) be a morphism of schemes. Consider the pullback functor [0BTI]\[\begin{equation} \text{schemes }U\text{ \'etale over }S \longrightarrow \begin{matrix} \text{descent data }(V, \varphi)\text{ relative to }X/S \\ \text{ with }V\text{ \'etale over }X \end{matrix} \end{equation}\] sending \(U\) to the canonical descent datum \((X \times_S U, can)\).
Lemma
If \(f : X \to S\) is surjective, then the functor (0BTI) is faithful.
Proof
Let \(a, b : U_1 \to U_2\) be two morphisms between schemes étale over \(S\). Assume the base changes of \(a\) and \(b\) to \(X\) agree. We have to show that \(a = b\). By Proposition 024V 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 \(u\).
Lemma
Assume \(f : X \to S\) is submersive and any étale base change of \(f\) is submersive. Then the functor (0BTI) is fully faithful.
Proof
By Lemma 0BTJ the functor is faithful. Let \(U_1 \to S\) and \(U_2 \to S\) be étale 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 \(U_1 \to U_2\).
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.
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 \(U_2 \to S\) is separated (Schemes, Lemma 01KV).
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 01KS. On the other hand the graph is open for example because it is a section of an étale morphism (Proposition 024T). 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 étale morphism whose base change to \(X\) is an isomorphism. This means that \(\Gamma \to U_1\) is universally bijective, hence an isomorphism by Theorem 025G. Thus \(\Gamma\) is the graph of a morphism \(U_1 \to U_2\) and the base change of this morphism is \(a\) as desired.
Lemma
Let \(f : X \to S\) be a morphism of schemes. In the following cases the functor (0BTI) is fully faithful:
\(f\) is surjective and universally closed (e.g., finite, integral, or proper),
\(f\) is surjective and universally open (e.g., locally of finite presentation and flat, smooth, or etale),
\(f\) is surjective, quasi-compact, and flat.
Proof
This follows from Lemma 0BTK. For example a closed surjective map of topological spaces is submersive (Topology, Lemma 0AAU). Finite, integral, and proper morphisms are universally closed, see Morphisms, Lemmas 01WM and 01WN and Definition 01W1. On the other hand an open surjective map of topological spaces is submersive (Topology, Lemma 02YB). Flat locally finitely presented, smooth, and étale morphisms are universally open, see Morphisms, Lemmas 01UA, 056G, and 03WT. The case of surjective, quasi-compact, flat morphisms follows from Morphisms, Lemma 02JY.
Lemma
Let \(f : X \to S\) be a morphism of schemes. Let \((V, \varphi)\) be a descent datum relative to \(X/S\) with \(V \to X\) étale. Let \(S = \bigcup S_i\) be an open covering. Assume that
the pullback of the descent datum \((V, \varphi)\) to \(X \times_S S_i/S_i\) is effective,
the functor (0BTI) for \(X \times_S (S_i \cap S_j) \to (S_i \cap S_j)\) is fully faithful, and
the functor (0BTI) for \(X \times_S (S_i \cap S_j \cap S_k) \to (S_i \cap S_j \cap S_k)\) is faithful.
Then \((V, \varphi)\) is effective.
Proof
(Recall that pullbacks of descent data are defined in Descent, Definition 02VR.) 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 étale 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 02VY (which is essentially just a reformulation of Schemes, Section 01JA) 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 \(\varphi\).
Lemma
Let \((A, I)\) be a henselian pair. Let \(U \to \Spec(A)\) be a quasi-compact, separated, étale 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 \(Z\).
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 05K0. 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 09XI 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.
Proposition
Let \(f : X \to S\) be a surjective integral morphism. The functor (0BTI) induces an equivalence \[\begin{matrix} \text{schemes quasi-compact,}\\ \text{separated, \'etale over }S \end{matrix} \longrightarrow \begin{matrix} \text{descent data }(V, \varphi)\text{ relative to }X/S\text{ with}\\ V\text{ quasi-compact, separated, \'etale over }X \end{matrix}\]
Proof
By Lemma 0BTL the functor (0BTI) 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 étale. We can use Lemma 0BTM to see that it suffices to prove the effectivity Zariski locally on \(S\). In particular we may and do assume that \(S\) is affine.
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 0BTG. 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 01ZM 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 étale, see Limits, Lemma 01ZQ and 07RP. Thus we may assume that \(f\) is finite surjective and \(S\) affine of finite type over \(\mathbf{Z}\).
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 \(\xi\).
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 052A. Let \((V_Z, \varphi_Z)\) be the pullback of the descent datum to \(X_Z/Z\). By More on Morphisms, Lemma 02W8 this descent datum is effective and we obtain an étale 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 02KQ and 02KU). Thus after shrinking once more we may assume that \(U_Z \to Z\) is finite, see Morphisms, Lemma 02NW.
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 0AGU) and by More on Morphisms, Lemma 02W8 we have fpqc descent for separated étale 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.
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 étale 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 09ZL we can find a finite étale 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 09XK) 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 0BTN. 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 09ZL 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 01JA). 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.
Normal crossings divisors
Here is the definition.
Definition
Let \(X\) be a locally Noetherian scheme. A 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 \(\mathcal{O}_{X, p}\).
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 supported on a strict normal crossings divisor.
Lemma
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
\(D\) is a strict normal crossings divisor, and
\(D\) 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 \(X\).
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 0BI9. 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 00NQ 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.
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 00NU. 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 033M). 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 00NQ and 00NA). 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.
Lemma
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 \(Y\).
Proof
As \(f\) is flat the pullback is defined by Divisors, Lemma 02OO 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 0BI9. 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 00TT). 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 00ON 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.
Here is the definition of a normal crossings divisor.
Definition
Let \(X\) be a locally Noetherian scheme. A normal crossings divisor on \(X\) is an effective Cartier divisor \(D \subset X\) such that for every \(p \in D\) there exists an étale morphism \(U \to X\) with \(p\) in the image and \(D \times_X U\) a strict normal crossings divisor on \(U\).
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 étale cover \(\Spec(\mathbf{C}[x, y])\) we obtain \((x - iy)(x + iy) = 0\).
Lemma
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 \(Y\).
Proof
As \(f\) is flat the pullback is defined by Divisors, Lemma 02OO hence the statement makes sense. Let \(q \in f^*D\) map to \(p \in D\). Choose an étale 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 0BSF. Set \(V = Y \times_X U\). Then \(V \to Y\) is étale as a base change of \(U \to X\) (Morphisms, Lemma 02GO) and the pullback \(D \times_X V\) is a strict normal crossings divisor on \(V\) by Lemma 0CBP. Thus we have checked the condition of Definition 0BSF for \(q \in f^*D\) and we conclude.
Lemma
Let \(X\) be a locally Noetherian scheme. Let \(D \subset X\) be a closed subscheme. The following are equivalent
\(D\) is a normal crossings divisor in \(X\),
\(D\) 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 \(n\).
Proof
First we explain how to think about condition (2). The diagonal of an unramified morphism is open (Morphisms, Lemma 02GE). 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 02HE) we have a conormal sheaf \(\mathcal{C}_n = \mathcal{C}_{Z_n/X}\) of \(Z_n \to X\), see More on Morphisms, Definition 04F4.
Formation of normalization commutes with étale localization by More on Morphisms, Lemma 07TD. 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 étale locally (see More on Algebra, Lemma 0AH0, Descent, Lemma 02VM, More on Morphisms, Lemma 069N and Descent, Lemma 05B2).
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 0BIA and More on Morphisms, Lemma 069L). 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 063I) it follows in the same manner that \(Z_n \to X\) has the required properties. Some details omitted.
Assume (2). Let \(p \in D\). Since \(D^\nu \to D\) is unramified, it is finite (by Morphisms, Lemma 01WJ). Hence \(D^\nu \to X\) is finite unramified. By Lemma 04HJ and étale 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 069G and Divisors, Lemma 063L. 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 00NU 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.
Lemma
Let \(X\) be a locally Noetherian scheme. Let \(D \subset X\) be a closed subscheme. If \(X\) is J-2 or Nagata, then the following are equivalent
\(D\) is a normal crossings divisor in \(X\),
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.
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 étale 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 \(\mathcal{O}_{U, u}\).
To prove the implication in the other direction we will use the criterion of Lemma 0CBR. Observe that formation of the normalization \(D^\nu \to D\) commutes with strict henselization, see More on Morphisms, Lemma 0CBM. 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.
If \(X\) is Nagata, then \(D^\nu \to D\) is finite by Morphisms, Lemma 035S.
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 étale 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 06LN). 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 étale 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 02LA) and the proof is complete.