Smooth algebra explained

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 \to C/N , there exists a k-algebra map 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.

k[[t _{1,} \ldots, t _{n]]} is 0-smooth only when \operatorname{char}k = p > 0 and [k: k ^{p]} < inf ty (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 \to C/N that is continuous when C/N is given the discrete topology, there exists an A-algebra map 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,} \ldots, t _{n]]} and I = (t _{1,} \ldots, t _{n).} Then B is I-smooth over A.ak{m} . Then A is ak{m} -smooth over k if and only if A ⊗ _{k} k' is a regular ring for any finite extension field k' of k .

See also

• étale morphism
• formally smooth morphism
• Popescu's theorem

References

• Book: Matsumura, H. . Reid . M. . Commutative Ring Theory . Cambridge University Press . Cambridge Studies in Advanced Mathematics . 1989 . 978-0-521-36764-6 .