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 , there exists a k-algebra map v : A → 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.
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 ] ] is 0-smooth only when char k = p > 0 and [ k : k p ] < ∞ (i.e., k has a finite p-basis.)
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 that is continuous when C / N is given the discrete topology, there exists an A-algebra map v : B → 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 ] ] and I = ( t 1 , … , t n ) . Then B is I-smooth over A.
Let A be a noetherian local k-algebra with maximal ideal m . Then A is m -smooth over k if and only if A ⊗ k k ′ is a regular ring for any finite extension field k ′ of k.
