ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...

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

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

''A'' denoted gl dim ''A'', is a non-negative

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of

projective dimension In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizat ...

s of all ''A''-

modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...

. Global dimension is an important technical notion in the dimension theory of

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 ...

s. By a theorem of

Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...

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

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

Noetherian

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 those rings which are regular. Their global dimension coincides with the

Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally th ...

, whose definition is module-theoretic. When the ring ''A'' is

noncommutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

, one initially has to consider two versions of this notion, right global dimension that arises from consideration of the right , and left global dimension that arises from consideration of the left . 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 In abstract algebra, the weak dimension of a nonzero right module ''M'' over a ring ''R'' is the largest number ''n'' such that the Tor group \operatorname_n^R(M,N) is nonzero for some left ''R''-module ''N'' (or infinity if no largest such ''n ...

'', 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.

Examples

*Let ''A'' = ''K'' 'x''₁,...,''x''_''n''be the

ring of polynomials In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...

in ''n'' variables over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''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; see

Hilbert's syzygy theorem In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important open questions in invariant theory, and are at ...

. More generally, if ''R'' is a Noetherian ring of finite global dimension ''k'' and ''A'' = ''R'' 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 In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...

. * 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. For example

\mathbb

has global dimension one. * The first

Weyl algebra In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More prec ...

''A''₁ is a noncommutative Noetherian

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...

of global dimension one. *If a ring is right Noetherian, then the right global dimension is the same as the weak global dimension, and is at most the left global dimension. In particular if a ring is right and left Noetherian then the left and right global dimensions and the weak global dimension are all the same. *The

triangular matrix ring In algebra, a triangular matrix ring, also called a triangular ring, is a ring constructed from two rings and a bimodule. Definition If T and U are rings and M is a \left(U,T\right)-bimodule, then the triangular matrix ring R:=\left beginT&0\\M&U ...

\begin\mathbb Z&\mathbb Q \\0&\mathbb Q \end

has right global dimension 1, weak global dimension 1, but left global dimension 2. It is right Noetherian but not left Noetherian.

Alternative characterizations

The right global dimension of a ring ''A'' can be alternatively defined as: * the supremum of the set of projective dimensions of all

cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in s ...

right ''A''-modules; * the supremum of the set of projective dimensions of all

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...

right ''A''-modules; * the supremum of the

injective dimension In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule of ...

s of all right ''A''-modules; * when ''A'' is a commutative Noetherian

with

maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...

''m'', the

of the

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 ...

''A''/''m''. The left global dimension of ''A'' has analogous characterizations obtained by replacing "right" with "left" in the above list. 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

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

References

* . * * . * {{citation , last1=McConnell , first1=J. C. , last2=Robson , first2=J. C. , last3=Small , first3=Lance W. , date=2001 , title=Noncommutative Noetherian Rings , series= Graduate Studies in Mathematics , volume=30 , publisher=American Mathematical Society , isbn=0-8218-2169-5 , editor=Revised. Ring theory Module theory Homological algebra Dimension