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 $u:\; B\; \backslash to\; C/N$ that is continuous when $C/N$ is given the discrete topology, there exists an A-algebra map $v:\; B\; \backslash to\; C$ 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, $B\; =\; A[\backslash ![t\_1,\; \backslash ldots,\; t\_n]\backslash !]$ and $I\; =\; (t\_1,\; \backslash ldots,\; t\_n).$ Then B is I-smooth over A.

Let A be a noetherian local k-algebra with maximal ideal $\backslash mathfrak\{m\}$. Then A is $\backslash mathfrak\{m\}$-smooth over $k$ if and only if $A\; \backslash otimes\_k\; k\text{'}$ is a regular ring for any finite extension field $k\text{'}$ of $k$.{{harvnb|Matsumura|1989|loc=Theorem 28.7}}

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

{{reflist}}

• {{cite book | last=Matsumura | first=H. | translator-last=Reid | translator-first=M. | title=Commutative Ring Theory | publisher=Cambridge University Press | series=Cambridge Studies in Advanced Mathematics | year=1989 | isbn=978-0-521-36764-6 | url=https://books.google.com/books?id=yJwNrABugDEC }}

Category:Algebras

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