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 ${\displaystyle u:A\to C/N}$, there exists a k-algebra map ${\displaystyle v:A\to C}$ 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.

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 ${\displaystyle k[\![t_{1},\ldots ,t_{n}]\!]}$ is 0-smooth only when ${\displaystyle \operatorname {char} k=p>0}$ and ${\displaystyle [k:k^{p}]<\infty }$ (i.e., k has a finite p-basis.)[2]

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 ${\displaystyle u:B\to C/N}$ that is continuous when ${\displaystyle C/N}$ is given the discrete topology, there exists an A-algebra map ${\displaystyle v:B\to C}$ 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, ${\displaystyle B=A[\![t_{1},\ldots ,t_{n}]\!]}$ and ${\displaystyle I=(t_{1},\ldots ,t_{n}).}$ Then B is I-smooth over A.

Let A be a noetherian local k-algebra with maximal ideal ${\displaystyle {\mathfrak {m}}}$. Then A is ${\displaystyle {\mathfrak {m}}}$-smooth over k if and only if ${\displaystyle A\otimes _{k}k'}$ is a regular ring for any finite extension field ${\displaystyle k'}$ of k.[3]

See also

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

References

1. Matsumura 1986, Theorem 25.3 harvnb error: no target: CITEREFMatsumura1986 (help)
2. Matsumura 1986, pg. 215 harvnb error: no target: CITEREFMatsumura1986 (help)
3. Matsumura 1986, Theorem 28.7 harvnb error: no target: CITEREFMatsumura1986 (help)

• H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.