commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...

and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu, states: :Let ''A'' be a

Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...

and ''B'' a Noetherian

algebra Algebra () is one of the 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 mathematics. Elementary ...

over it. Then, the structure map ''A'' → ''B'' is a regular homomorphism

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...

''B'' is a direct limit of smooth ''A''-algebras. For example, if ''A'' is a

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 ...

G-ring In commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular (defined below). Almost all Noetherian rings that occur naturally in algebraic geometry or number ...

(e.g., a local excellent ring) and ''B'' its completion, then the map ''A'' → ''B'' is regular by definition and the theorem applies. Another proof of Popescu's theorem was given by Tetsushi Ogoma, while an exposition of the result was provided by

Richard Swan Richard Gordon Swan (; born 1933) is an American mathematician who is known for the Serre–Swan theorem relating the geometric notion of vector bundles to the algebraic concept of projective modules, and for the Swan representation, an ''l''-a ...

. The usual proof of the

Artin approximation theorem In mathematics, the Artin approximation theorem is a fundamental result of in deformation theory which implies that formal power series with coefficients in a field (mathematics), field ''k'' are well-approximated by the algebraic functions on ''k' ...

relies crucially on Popescu's theorem. Popescu's result was proved by an alternate method, and somewhat strengthened, by Mark Spivakovsky.

References

External links

* {{DEFAULTSORT:Popescu's theorem Theorems in algebraic geometry

See also

References

External links