The Stacks project

41.3 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 10.151) and it is different. After discussing the notion a bit we globalize it to describe unramified morphisms of locally Noetherian schemes.

Definition 41.3.1. Let $A$, $B$ be Noetherian local rings. A local homomorphism $A \to B$ is said to be unramified homomorphism of local rings if

  1. $\mathfrak m_ AB = \mathfrak m_ B$,

  2. $\kappa (\mathfrak m_ B)$ is a finite separable extension of $\kappa (\mathfrak m_ A)$, and

  3. $B$ 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 10.151. 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 10.151.5 and 10.151.7 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.

slogan

Lemma 41.3.2. 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. $\square$

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 10.97.6.

Lemma 41.3.3. Let $A$, $B$ be Noetherian local rings. Let $A \to B$ be a local homomorphism.

  1. if $A \to B$ is an unramified homomorphism of local rings, then $B^\wedge $ is a finite $A^\wedge $ module,

  2. 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,

  3. 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,

  4. 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 10.97.7. 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 10.20.1) 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. $\square$

Lemma 41.3.4. 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

  1. $A \to B$ is an unramified homomorphism of local rings

  2. $A^\wedge \to B^\wedge $ is an unramified homomorphism of local rings, and

  3. $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 41.3.3 we see that $A^\wedge \to B^\wedge $ is finite. Hence $A^\wedge \to B^\wedge $ is of finite presentation, and by Algebra, Lemma 10.151.7 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 10.151.5 we conclude that $A^\wedge \to B^\wedge $ is an unramified homomorphism of local rings, i.e., (2) holds. $\square$

Definition 41.3.5. (See Morphisms, Definition 29.35.1 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 $x \in X$.

  1. 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.

  2. 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 41.3.6. 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 41.3.5 if and only if it is unramified in the sense of Morphisms, Definition 29.35.1.

Proof. This follows from Lemma 41.3.2 and the definitions. $\square$

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:

  1. Unramifiedness is local on the source and the target in the Zariski topology.

  2. Unramified morphisms are stable under base change and composition. See Morphisms, Lemmas 29.35.5 and 29.35.4.

  3. Unramified morphisms of schemes are locally quasi-finite and quasi-compact unramified morphisms are quasi-finite. See Morphisms, Lemma 29.35.10

  4. Unramified morphisms have relative dimension $0$. See Morphisms, Definition 29.29.1 and Morphisms, Lemma 29.29.5.

  5. 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 29.35.12.

  6. Let $X$ and $Y$ be unramified over a base scheme $S$. Any $S$-morphism from $X$ to $Y$ is unramified. See Morphisms, Lemma 29.35.16.

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