In mathematics, the Cohen structure theorem, introduced by , describes the structure of
complete Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
s.
Some consequences of Cohen's structure theorem include three conjectures of
Krull:
*Any complete
regular equicharacteristic Noetherian local ring is a
ring of formal power series
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 sum ...
over a field. (Equicharacteristic means that the local ring and its
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
have the same
characteristic, and is equivalent to the local ring containing a field.)
*Any complete regular Noetherian local ring that is not equicharacteristic but is unramified is uniquely determined by its residue field and its dimension.
*Any complete Noetherian local ring is the image of a complete regular Noetherian local ring.
Statement
The most commonly used case of Cohen's theorem is when the complete Noetherian local ring contains some field. In this case Cohen's structure theorem states that the ring is of the form ''k''
''x''1,...,''x''''n''/(''I'') for some ideal ''I'', where ''k'' is its residue class field.
In the unequal characteristic case when the complete Noetherian local ring does not contain a field, Cohen's structure theorem states that the local ring is a quotient of a formal power series ring in a finite number of variables over a
Cohen ring
In algebra, a Cohen ring is a field or a complete discrete valuation ring of mixed characteristic (0, p) whose maximal ideal is generated by ''p''. Cohen rings are used in the Cohen structure theorem for complete Noetherian local ring In abstr ...
with the same residue field as the local ring. A Cohen ring is a field or a complete characteristic zero
discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R'' i ...
whose maximal ideal is generated by a prime number ''p'' (equal to the characteristic of the residue field).
In both cases, the hardest part of Cohen's proof is to show that the complete Noetherian local ring contains a coefficient ring (or coefficient field), meaning a complete discrete valuation ring (or field) with the same residue field as the local ring.
All this material is developed carefully in the Stacks Project .
References
* Cohen's paper was written when "local ring" meant what is now called a "Noetherian local ring".
*{{Citation , last1=Samuel , first1=Pierre , author1-link=Pierre Samuel , title=Algèbre locale , url=https://books.google.com/books?id=enNFAAAAYAAJ , publisher=Gauthier-Villars , series=Mémor. Sci. Math. , mr=0054995 , year=1953 , volume=123
Commutative algebra
Theorems in ring theory