In
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. Promi ...
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 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 ...
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 bi ...
''B'' is a
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cat ...
of
smooth ''A''-algebras.
For example, if ''A'' is a
local G-ring (e.g., a local
excellent ring In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the ...
) and ''B'' its
completion, then the map ''A'' → ''B'' is regular by definition and the theorem applies.
Another
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a con ...
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''- ...
.
The usual proof of the
Artin approximation theorem relies crucially on Popescu's theorem. Popescu's result was proved by an alternate method, and somewhat strengthened, by Mark Spivakovsky.
See also
*
Ring with the approximation property
In algebra, a commutative Noetherian ring ''A'' is said to have the approximation property with respect to an ideal ''I'' if each finite system of polynomial equations with coefficients in ''A'' has a solution in ''A'' if and only if it has a solut ...
References
External links
*
{{DEFAULTSORT:Popescu's theorem
Theorems in algebraic geometry