Lemma 41.19.1. Let $A$, $B$ be Noetherian local rings. Let $A \to B$ be a étale homomorphism of local rings. Then $\dim (A) = \dim (B)$.
41.19 Permanence properties
In what follows, we present a few “permanence” properties of étale homomorphisms of Noetherian local rings (as defined in Definition 41.11.1). See More on Algebra, Sections 15.43 and 15.45 for the analogue of this material for the completion and henselization of a Noetherian local ring.
Proof. See for example Algebra, Lemma 10.112.7. $\square$
Proposition 41.19.2. 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 10.163.2. $\square$
Proposition 41.19.3. 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. $\square$
Proposition 41.19.4. 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 10.110.9. 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 41.19.1). 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 10.112.8.) $\square$
Proposition 41.19.5. 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 10.164.2. 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 41.11.2. Then we see that Algebra, Lemma 10.163.7 applies to $A \to B'$ and $B'$ is reduced. Hence $B$ is reduced. $\square$
Remark 41.19.6. 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 110.17. 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 41.19.7. 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 10.164.3 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 41.11.2. Then we see that Algebra, Lemma 10.163.9 applies to $A \to B'$ and we conclude that $B'$ is normal. Hence $B$ is a normal domain. $\square$
The preceding propositions give some indication as to why we'd like to think of étale maps as “local isomorphisms”. Another property that gives an excellent indication that we have the “right” definition is the fact that for $\mathbf{C}$-schemes of finite type, a morphism is étale if and only if the associated morphism on analytic spaces (the $\mathbf{C}$-valued points given the complex topology) is a local isomorphism in the analytic sense (open embedding locally on the source). This fact can be proven with the aid of the structure theorem and the fact that the analytification commutes with the formation of the completed local rings – the details are left to the reader.
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