In algebra, a commutative k-algebra A is said to be 0-smooth if it satisfies the following lifting property: given a k-algebra C, an ideal N of C whose square is zero and a k-algebra map u : A → C / N {\displaystyle u:A\to C/N} , there exists a k-algebra map v : A → C {\displaystyle v:A\to C} such that u is v followed by the canonical map. If there exists at most one such lifting v, then A is said to be 0-unramified (or 0-neat). A is said to be 0-étale if it is 0-smooth and 0-unramified. The notion of 0-smoothness is also called formal smoothness. A finitely generated k-algebra A is 0-smooth over k if and only if Spec A is a smooth scheme over k. A separable algebraic field extension L of k is 0-étale over k. The formal power series ring k [ [ t 1 , … , t n ] ] {\displaystyle k[\![t_{1},\ldots ,t_{n}]\!]} is 0-smooth only when char k = p > 0 {\displaystyle \operatorname {char} k=p>0} and [ k : k p ] < ∞ {\displaystyle [k:k^{p}]<\infty } (i.e., k has a finite p-basis.)
I-smooth Let B be an A-algebra and suppose B is given the I-adic topology, I an ideal of B. We say B is I-smooth over A if it satisfies the lifting property: given an A-algebra C, an ideal N of C whose square is zero and an A-algebra map u : B → C / N {\displaystyle u:B\to C/N} that is continuous when C / N {\displaystyle C/N} is given the discrete topology, there exists an A-algebra map v : B → C {\displaystyle v:B\to C} such that u is v followed by the canonical map. As before, if there exists at most one such lift v, then B is said to be I-unramified over A (or I-neat). B is said to be I-étale if it is I-smooth and I-unramified. If I is the zero ideal and A is a field, these notions coincide with 0-smooth etc. as defined above. A standard example is this: let A be a ring, B = A [ [ t 1 , … , t n ] ] {\displaystyle B=A[\![t_{1},\ldots ,t_{n}]\!]} and I = ( t 1 , … , t n ) . {\displaystyle I=(t_{1},\ldots ,t_{n}).} Then B is I-smooth over A. Let A be a noetherian local k-algebra with maximal ideal m {\displaystyle {\mathfrak {m}}} . Then A is m {\displaystyle {\mathfrak {m}}} -smooth over k {\displaystyle k} if and only if A ⊗ k k ′ {\displaystyle A\otimes _{k}k'} is a regular ring for any finite extension field k ′ {\displaystyle k'} of k {\displaystyle k} .
See also étale morphism formally smooth morphism Popescu's theorem
Notes
References Matsumura, H. (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Translated by Reid, M. Cambridge University Press. ISBN 978-0-521-36764-6.