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

A separable algebraic field extension L of k is 0-étale over k. The formal power series ring [math]\displaystyle{ k[\![t_1, \ldots, t_n]\!] }[/math] is 0-smooth only when [math]\displaystyle{ \operatorname{char}k = p \gt 0 }[/math] and [math]\displaystyle{ [k: k^p] \lt \infty }[/math] (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 [math]\displaystyle{ u: B \to C/N }[/math] that is continuous when [math]\displaystyle{ C/N }[/math] is given the discrete topology, there exists an A-algebra map [math]\displaystyle{ v: B \to C }[/math] 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, [math]\displaystyle{ B = A[\![t_1, \ldots, t_n]\!] }[/math] and [math]\displaystyle{ I = (t_1, \ldots, t_n). }[/math] Then B is I-smooth over A.

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

See also

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

References

• Matsumura, H. (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press. ISBN 978-0-521-36764-6. https://books.google.com/books?id=yJwNrABugDEC.