# Global dimension > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Global_dimension > Markdown URL: https://mediated.wiki/source/Global_dimension.md > Source: https://en.wikipedia.org/wiki/Global_dimension > Source revision: 1262791737 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Concept in ring theory and homological algebra}} In [ring theory](/source/ring_theory) and [homological algebra](/source/homological_algebra), the '''global dimension''' (or '''global homological dimension'''; sometimes just called '''homological dimension''') of a [ring](/source/ring_(mathematics)) ''A'' denoted gl dim ''A'', is a non-negative [integer](/source/integer) or infinity which is a homological invariant of the ring. It is defined to be the [supremum](/source/supremum) of the set of [projective dimension](/source/projective_dimension)s of all ''A''-[modules](/source/module_(mathematics)). Global dimension is an important technical notion in the dimension theory of [Noetherian ring](/source/Noetherian_ring)s. By a theorem of [Jean-Pierre Serre](/source/Jean-Pierre_Serre), global dimension can be used to characterize within the class of [commutative](/source/commutative_ring) Noetherian [local ring](/source/local_ring)s those rings which are [regular](/source/regular_local_ring). Their global dimension coincides with the [Krull dimension](/source/Krull_dimension), whose definition is module-theoretic. When the ring ''A'' is [noncommutative](/source/noncommutative_ring), one initially has to consider two versions of this notion, right global dimension that arises from consideration of the right {{nowrap|''A''-modules}}, and left global dimension that arises from consideration of the left {{nowrap|''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](/source/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.{{cite journal|last1=Auslander|first1=Maurice|title=On the dimension of modules and algebras. III. Global dimension|journal=Nagoya Math J.|date=1955|volume=9|pages=67–77|doi=10.1017/S0027763000023291 |url=http://projecteuclid.org/euclid.nmj/1118799684|ref=AuslanderGlobalDimension}} == Examples == *Let ''A'' = ''K''[''x''1,...,''x''''n''] be the [ring of polynomials](/source/ring_of_polynomials) in ''n'' variables over a [field](/source/field_(mathematics)) ''K''. Then the global dimension of ''A'' is equal to ''n''. This statement goes back to [David Hilbert](/source/David_Hilbert)'s foundational work on homological properties of polynomial rings; see [Hilbert's syzygy theorem](/source/Hilbert'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](/source/semisimple_ring). * The global dimension of a ring ''A'' is less than or equal to one if and only if ''A'' is [hereditary](/source/hereditary_ring). In particular, a commutative [principal ideal domain](/source/principal_ideal_domain) which is not a field has global dimension one. For example \mathbb{Z} has global dimension one. * The first [Weyl algebra](/source/Weyl_algebra) ''A''1 is a noncommutative Noetherian [domain](/source/domain_(ring_theory)) 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](/source/triangular_matrix_ring) \begin{bmatrix}\mathbb Z&\mathbb Q \\0&\mathbb Q \end{bmatrix} 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](/source/cyclic_module) right ''A''-modules; * the supremum of the set of projective dimensions of all [finite](/source/finitely-generated_module) right ''A''-modules; * the supremum of the [injective dimension](/source/injective_dimension)s of all right ''A''-modules; * when ''A'' is a commutative Noetherian [local ring](/source/local_ring) with [maximal ideal](/source/maximal_ideal) ''m'', the [projective dimension](/source/projective_dimension) of the [residue field](/source/residue_field) ''A''/''m''. The left global dimension of ''A'' has analogous characterizations obtained by replacing "right" with "left" in the above list. Serre [proved](/source/mathematical_proof) that a commutative Noetherian local ring ''A'' is [regular](/source/regular_local_ring) if and only if it has finite global dimension, in which case the global dimension coincides with the [Krull dimension](/source/Krull_dimension) of ''A''. This theorem opened the door to application of homological methods to commutative algebra. == References == {{reflist}} * {{citation | last=Eisenbud | first=David | author-link=David Eisenbud | title=Commutative Algebra with a View Toward Algebraic Geometry | series=Graduate Texts in Mathematics | volume=150 | publisher=Springer-Verlag | date=1999 | edition=3rd | isbn=0-387-94268-8}}. * {{citation | last=Kaplansky | first=Irving | authorlink=Irving Kaplansky | title = Fields and Rings | edition=2nd | zbl=1001.16500 | series=Chicago Lectures in Mathematics | publisher = University Of Chicago Press | year = 1972 | isbn = 0-226-42451-0 }} * {{citation | last=Matsumura | first=Hideyuki | title=Commutative Ring Theory | series=Cambridge Studies in Advanced Mathematics | volume=8 | date=1989 | publisher=Cambridge University Press | isbn=0-521-36764-6}}. * {{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](/source/Graduate_Studies_in_Mathematics) | volume=30 | publisher=American Mathematical Society | isbn=0-8218-2169-5 | editor=Revised}}. Category:Ring theory Category:Module theory Category:Homological algebra Category:Dimension --- Adapted from the Wikipedia article [Global dimension](https://en.wikipedia.org/wiki/Global_dimension) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Global_dimension?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.