In ring theory

Ring theory

In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...

 and homological algebra

Homological algebra

Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...

, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

 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 the supremum

Supremum

In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...

 of the set of projective dimensions of all A-modules

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

. Global dimension is an important technical notion in the dimension theory of Noetherian rings. By a theorem of Jean-Pierre Serre

Jean-Pierre Serre

Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...

, global dimension can be used to characterize within the class of commutative Noetherian

Noetherian ring

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...

 local ring

Local ring

In abstract algebra, more particularly in 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 number fields examined at a particular place, or...

s those rings which are regular

Regular local ring

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of...

. Their global dimension coincides with the Krull dimension

Krull dimension

In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....

, 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

Noetherian ring

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...

, 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 first Weyl algebra A₁ is a noncommutative Noetherian domain

Domain (ring theory)

In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. Some authors require the ring to be nontrivial...

 of global dimension one.

Examples

Let A = K[x₁,...,x_n] be the ring of polynomials

Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...

 in n variables over a field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 K. Then the global dimension of A is equal to n. This statement goes back to David Hilbert

David Hilbert

David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

's foundational work on homological properties of polynomial rings, see Hilbert's syzygy theorem

Hilbert's syzygy theorem

In mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert in connection with the syzygy problem of invariant theory. Roughly speaking, starting with relations between polynomial invariants, then relations between the relations, and so on, it...

. 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

Hereditary ring

In mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective...

. In particular, a commutative principal ideal domain

Principal ideal domain

In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...

 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
  Cyclic module
  In mathematics, more specifically in ring theory, a cyclic module is a module over a ring which is generated by one element. The term is by analogy with cyclic groups, that is groups which are generated by one element.- Definition :...
  
   A-modules;
• the supremum of the set of projective dimensions of all finite
  Finitely-generated module
  In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated R-module also may be called a finite R-module or finite over R....
  
   A-modules;
• the supremum of the injective dimensions of all A-modules;
• when A is a commutative
  Commutative ring
  In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
  
   Noetherian
  Noetherian ring
  In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
  
   local ring
  Local ring
  In abstract algebra, more particularly in 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 number fields examined at a particular place, or...
  
   with maximal ideal
  Maximal ideal
  In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
  
   m, the projective dimension of the residue field
  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...
  
   A/m.

Serre

Jean-Pierre Serre

Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...

 proved that a commutative Noetherian local ring A is regular

Regular local ring

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of...

 if and only if it has finite global dimension, in which case the global dimension coincides with the Krull dimension

Krull dimension

In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....

of A. This theorem opened the door to application of homological methods to commutative algebra.