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.
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.
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
Comments (2)
Comment #864 by Keenan Kidwell on
Comment #871 by Johan on