{{one source |date=May 2024}} 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>u: A \to C/N</math>, there exists a ''k''-algebra map <math>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''.<ref>{{harvnb|Matsumura|1989|loc=Theorem 25.3}}</ref> The formal power series ring <math>k[\![t_1, \ldots, t_n]\!]</math> is 0-smooth only when <math>\operatorname{char}k = p > 0</math> and <math>[k: k^p] < \infty</math> (i.e., ''k'' has a finite ''p''-basis.)<ref>{{harvnb|Matsumura|1989|loc=pg. 215}}</ref>
== ''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>u: B \to C/N</math> that is continuous when <math>C/N</math> is given the discrete topology, there exists an ''A''-algebra map <math>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>B = A[\![t_1, \ldots, t_n]\!]</math> and <math>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>\mathfrak{m}</math>. Then ''A'' is <math>\mathfrak{m}</math>-smooth over <math>k</math> if and only if <math>A \otimes_k k'</math> is a regular ring for any finite extension field <math>k'</math> of <math>k</math>.<ref>{{harvnb|Matsumura|1989|loc=Theorem 28.7}}</ref>
== See also == *étale morphism *formally smooth morphism *Popescu's theorem
==Notes== {{reflist}}
== References == * {{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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