Smooth algebra

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 ${\displaystyle k}$ if and only if ${\displaystyle A\otimes _{k}k'}$ is a regular ring for any finite extension field ${\displaystyle k'}$ of ${\displaystyle k}$.[3]

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

• Matsumura, H. (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Translated by Reid, M. Cambridge University Press. ISBN 978-0-521-36764-6.