- Global dimension
In
ring theory andhomological algebra , the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring "A" denoted gl dim "A", is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be thesupremum of the set ofprojective dimension s of all "A"-modules. Global dimension is an important technical notion in the dimension theory of Noetherian rings. By a theorem ofJean-Pierre Serre , global dimension can be used to characterize within the class of commutativeNoetherian local ring s those rings which are regular. Their global dimension coincides with theKrull dimension , whose definition is module-theoretic.When the ring "A" is noncommutative, one initially has to consider two versions of this notion, right global dimension that arises from consideration of the right "A"-modules, and left global dimension that arises from consideration of the left "A"-modules. For an arbitrary ring "A" the right and left global dimensions may differ. However, if "A" is a
Noetherian ring , both of these dimensions turn out to be equal to weak global dimension, whose definition is left-right symmetric. Therefore, for noncommutative Noetherian rings, these two versions coincide and one is justified in talking about the global dimension. The firstWeyl algebra "A""1" is a noncommutative Noetherian domain of global dimension one.Examples
Let "A" = K ["x""1",...,"x""n"] be the ring of polynomials in "n" variables over a field K. Then the global dimension of "A" is equal to "n". This statement goes back to
David Hilbert 's foundational work on homological properties of polynomial rings, seeHilbert's syzygy theorem . More generally, if "R" is a Noetherian ring of finite global dimension "k" and "A" = "R" [x] is a ring of polynomials in one variable over "R" then the global dimension of "A" is equal to "k" + 1.A ring has global dimension zero if and only if it is semisimple. The global dimension of a ring "A" is less than or equal to one if and only if "A" is hereditary. In particular, a commutative
principal ideal domain which is not a field has global dimension one.Alternative characterizations
The global dimension of a ring "A" can be alternatively defined as:
* the supremum of the set of projective dimensions of all cyclic "A"-modules;
* the supremum of the set of projective dimensions of all finite "A"-modules;
* the supremum of theinjective dimension s of all "A"-modules;
* when "A" is a commutativeNoetherian local ring withmaximal ideal "m", theprojective dimension of theresidue field "A"/"m".Serre proved that a commutative Noetherian local ring "A" is regular if and only if it has finite global dimension, in which case the global dimension coincides with the
Krull dimension of "A". This theorem opened the door to application of homological methods to commutative algebra.References
*.
*.
*.
Wikimedia Foundation. 2010.
