Definition 10.150.1. Let $R \to S$ be a ring map. We say $S$ is formally étale over $R$ if for every commutative solid diagram where $I \subset A$ is an ideal of square zero, there exists a unique dotted arrow making the diagram commute.
\[ \xymatrix{ S \ar[r] \ar@{-->}[rd] & A/I \\ R \ar[r] \ar[u] & A \ar[u] } \]
Clearly a ring map is formally étale if and only if it is both formally smooth and formally unramified.
Lemma 10.150.2. Let $R \to S$ be a ring map of finite presentation. The following are equivalent:
$R \to S$ is formally étale,
$R \to S$ is étale.
Proof. Assume that $R \to S$ is formally étale. Then $R \to S$ is smooth by Proposition 10.138.13. By Lemma 10.148.2 we have $\Omega _{S/R} = 0$. Hence $R \to S$ is étale by definition.
Assume that $R \to S$ is étale. Then $R \to S$ is formally smooth by Proposition 10.138.13. By Lemma 10.148.2 it is formally unramified. Hence $R \to S$ is formally étale. $\square$
Lemma 10.150.3. Let $R$ be a ring. Let $I$ be a directed set. Let $(S_ i, \varphi _{ii'})$ be a system of $R$-algebras over $I$. If each $R \to S_ i$ is formally étale, then $S = \mathop{\mathrm{colim}}\nolimits _{i \in I} S_ i$ is formally étale over $R$
Proof. Consider a diagram as in Definition 10.150.1. By assumption we get unique $R$-algebra maps $S_ i \to A$ lifting the compositions $S_ i \to S \to A/I$. Hence these are compatible with the transition maps $\varphi _{ii'}$ and define a lift $S \to A$. This proves existence. The uniqueness is clear by restricting to each $S_ i$. $\square$
Lemma 10.150.4. Let $R$ be a ring. Let $S \subset R$ be any multiplicative subset. Then the ring map $R \to S^{-1}R$ is formally étale.
Proof. Let $I \subset A$ be an ideal of square zero. What we are saying here is that given a ring map $\varphi : R \to A$ such that $\varphi (f) \mod I$ is invertible for all $f \in S$ we have also that $\varphi (f)$ is invertible in $A$ for all $f \in S$. This is true because $A^*$ is the inverse image of $(A/I)^*$ under the canonical map $A \to A/I$. $\square$
Lemma 10.150.5. Let $R \to S$ be a ring map. Let $J \subset S$ be an ideal such that $R \to S/J$ is surjective; let $I \subset R$ be the kernel. If $R \to S$ is formally étale, then $\bigoplus I^ n/I^{n + 1} \to \bigoplus J^ n/J^{n + 1}$ is an isomorphism of graded rings.
Proof. Using the lifting property inductively we find dotted arrows
The corresponding maps $S/J^ n \to R/I^ n$ are isomorphisms since the compositions $S/J^ n \to R/I^ n \to S/J^ n$ are (inductively) the identity by the uniqueness in the lifting property of formally étale ring maps. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (2)
Comment #864 by Keenan Kidwell on
Comment #871 by Johan on