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 (or 0-neat).

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.[1] The formal power series ring

(i.e., k has a finite p-basis.

)[2] 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 (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.

A standard example is this: let A be a ring,

Let A be a noetherian local k-algebra with maximal ideal

is a regular ring for any finite extension field

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