Basic solution (linear programming): Difference between revisions
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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 a 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'''. | |||
A separable algebraic field extension ''L'' of ''k'' is 0-étale over ''k''.<ref>{{harvnb|Matsumura|1986|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</math> and <math>[k: k^p] < \infty</math> (i.e., ''k'' has a finite [[p-basis|''p''-basis]].)<ref>{{harvnb|Matsumura|1986|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 ring with maximal ideal <math>\mathfrak{m}</math> that is a ''k''-algebra. Then ''A'' is <math>\mathfrak{m}</math>-smooth over ''k'' if and only if <math>A \otimes_k k'</math> is a regular local ring for any finite field extension <math>k'</math> of ''k''.<ref>{{harvnb|Matsumura|1986|loc=Theorem 28.7}}</ref> | |||
== See also == | |||
*[[étale morphism]] | |||
*[[formally smooth morphism]] | |||
== References == | |||
{{reflist}} | |||
* H. Matsumura ''Commutative ring theory.'' Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. | |||
[[Category:Algebra]] | |||
{{algebra-stub}} | |||
Revision as of 14:05, 24 January 2014
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 , there exists a k-algebra map such that u is v followed by the canonical map. If there exists at most one such a 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.
A separable algebraic field extension L of k is 0-étale over k.[1] the formal power series ring is 0-smooth only when and (i.e., k has a finite p-basis.)[2]
![{\displaystyle k[\![t_{1},\ldots ,t_{n}]\!]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce3f2737045a1caafcd0b26c3851c2bbf492ffd)
![{\displaystyle [k:k^{p}]<\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7030d6dbe7ecf3ead7fbc03dc1a9f7b85f3ad86d)
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 that is continuous when is given the discrete topology, there exists an A-algebra map 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, and Then B is I-smooth over A.
![{\displaystyle B=A[\![t_{1},\ldots ,t_{n}]\!]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d81bb1e134c4ace0c8adbb8059b550ecb1d86b79)
Let A be a noetherian local ring with maximal ideal that is a k-algebra. Then A is -smooth over k if and only if is a regular local ring for any finite field extension of k.[3]
See also
References
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- H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.