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 → C / N {\displaystyle u:A\to C/N} , there exists a k-algebra map v : A → C {\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. 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 k [ [ t 1 , … , t n ] ] {\displaystyle k[\![t_{1},\ldots ,t_{n}]\!]} is 0-smooth only when char k = p > 0 {\displaystyle \operatorname {char} k=p>0} and [ k : k p ] < ∞ {\displaystyle [k:k^{p}]<\infty } (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 → C / N {\displaystyle u:B\to C/N} that is continuous when C / N {\displaystyle C/N} is given the discrete topology, there exists an A-algebra map v : B → C {\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, B = A [ [ t 1 , … , t n ] ] {\displaystyle B=A[\![t_{1},\ldots ,t_{n}]\!]} and I = ( t 1 , … , t n ) . {\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 m {\displaystyle {\mathfrak {m}}} . Then A is m {\displaystyle {\mathfrak {m}}} -smooth over k {\displaystyle k} if and only if A ⊗ k k ′ {\displaystyle A\otimes _{k}k'} is a regular ring for any finite extension field k ′ {\displaystyle k'} of k {\displaystyle k} .3
See also
Notes
- Matsumura, H. (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Translated by Reid, M. Cambridge University Press. ISBN 978-0-521-36764-6.
References
Matsumura 1989, Theorem 25.3 - Matsumura, H. (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Translated by Reid, M. Cambridge University Press. ISBN 978-0-521-36764-6. https://books.google.com/books?id=yJwNrABugDEC ↩
Matsumura 1989, pg. 215 - Matsumura, H. (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Translated by Reid, M. Cambridge University Press. ISBN 978-0-521-36764-6. https://books.google.com/books?id=yJwNrABugDEC ↩
Matsumura 1989, Theorem 28.7 - Matsumura, H. (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Translated by Reid, M. Cambridge University Press. ISBN 978-0-521-36764-6. https://books.google.com/books?id=yJwNrABugDEC ↩