41.12 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 10.144.1. Namely, suppose that $R$ is a ring and $f, g \in R[t]$ are polynomials such that

1. $f$ is a monic polynomial, and

2. $f' = \text{d}f/\text{d}t$ 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 41.12.1. Let $f : A \to B$ be an étale homomorphism of local rings. Then there exist $f, g \in A[t]$ such that

1. $B' = A[t]_ g/(f)$ is standard étale – see (a) and (b) above, and

2. $B$ 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 41.11.2 we see that $A \to B'$ is étale at $\mathfrak q$. Hence we may apply Algebra, Proposition 10.144.4 to see that a principal localization of $B'$ is standard étale. $\square$

Here is the version for unramified homomorphisms of local rings.

Theorem 41.12.2. Let $f : A \to B$ be an unramified morphism of local rings. Then there exist $f, g \in A[t]$ such that

1. $B' = A[t]_ g/(f)$ is standard étale – see (a) and (b) above, and

2. $B$ 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 41.3.2 we see that $A \to B'$ is unramified at $\mathfrak q$. Hence we may apply Algebra, Proposition 10.152.1 to see that a principal localization of $B'$ is a quotient of a standard étale $A$-algebra. $\square$

Via standard lifting arguments, one then obtains the following geometric statement which will be of essential use to us.

Theorem 41.12.3. 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

$\xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-j & \mathop{\mathrm{Spec}}(R[t]_{f'}/(f)) \ar[d] \\ Y & V \ar[l] \ar@{=}[r] & \mathop{\mathrm{Spec}}(R) }$

where $j$ is an open immersion, and $f \in R[t]$ is monic.

Proof. This is equivalent to Morphisms, Lemma 29.36.14 although the statements differ slightly. See also, Varieties, Lemma 33.18.3 for a variant for unramified morphisms. $\square$

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