Smooth algebra

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, there exists a k-algebra map 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. 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 is 0-smooth only when and

''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 that is continuous when is given the discrete topology, there exists an A-algebra map 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. 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, and Then B is I-smooth over A.
Let A be a noetherian local k-algebra with maximal ideal. Then A is -smooth over if and only if is a regular ring for any finite extension field of.