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In
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
, a commutative ''k''-algebra ''A'' is said to be 0-smooth if it satisfies the following lifting property: given a ''k''-
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
''C'', an ideal ''N'' of ''C'' whose square is
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...
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 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 In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a s ...
over ''k''. A separable algebraic
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
''L'' of ''k'' is 0-étale over ''k''. The
formal power series ring In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
k ![t_1, \ldots, t_n!">_1,_\ldots,_t_n.html" ;"title="![t_1, \ldots, t_n">![t_1, \ldots, t_n!/math> is 0-smooth only when \operatornamek = p > 0 and [k: k^p] < \infty (i.e., ''k'' has a finite p-basis, ''p''-basis.)


''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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
when C/N is given the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest t ...
, 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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, these notions coincide with 0-smooth etc. as defined above. A standard example is this: let ''A'' be a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, B = A ![t_1, \ldots, t_n!">_1,_\ldots,_t_n.html" ;"title="![t_1, \ldots, t_n">![t_1, \ldots, t_n!/math> and I = (t_1, \ldots, t_n). Then ''B'' is ''I''-smooth over ''A''. Let ''A'' be a noetherian local ring">local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administrat ...
''k''-algebra with maximal ideal \mathfrak. Then ''A'' is \mathfrak-smooth over k if and only if A \otimes_k k' is a regular ring for any finite extension, finite extension field k' of k.


See also

*
étale morphism In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy ...
*
formally smooth morphism In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if *(i) it is locally of finite presentation *(ii) it is flat, and *(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular. (iii) means ...
*
Popescu's theorem In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu, states: :Let ''A'' be a Noetherian ring and ''B'' a Noetherian algebra over it. Then, the structure map ''A'' → ''B'' is a regular homomorphism if and ...


References

* Algebra {{algebra-stub