Basic solution (linear programming): Difference between revisions

Revision as of 13:05, 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 $u : A \to C / N$ , there exists a k-algebra map $v : A \to C$ 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 $k [[t_{1}, \dots, t_{n}]]$ is 0-smooth only when $char k = p$ and $[k : k^{p}] < \infty$ (i.e., k has a finite p-basis.)^[2]

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 $u : B \to C / N$ that is continuous when $C / N$ is given the discrete topology, there exists an A-algebra map $v : B \to C$ 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, $B = A [[t_{1}, \dots, t_{n}]]$ and $I = (t_{1}, \dots, t_{n}) .$ Then B is I-smooth over A.

Let A be a noetherian local ring with maximal ideal $m$ that is a k-algebra. Then A is $m$ -smooth over k if and only if $A \otimes_{k} k^{'}$ is a regular local ring for any finite field extension $k^{'}$ of k.^[3]

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.

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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}}

Basic solution (linear programming): Difference between revisions

Revision as of 13:05, 24 January 2014

I-smooth

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