# Global dimension

The **global dimension** of a ring \( R \), written \( \operatorname{gl\,dim} R \), is the supremum of the projective dimensions of all (left) \( R \)-modules, that is, the smallest length \( n \) such that every module admits a projective resolution of length at most \( n \).<sup>[1](https://encyclopediaofmath.org/wiki/Dimension)</sup> It measures how far the module category of \( R \) is from being resolved by projective modules, and it is also called the homological dimension of the ring.<sup>[2](https://planetmath.org/globaldimension)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( \operatorname{gl\,dim} R = \sup\{\operatorname{pd}_R(M) : M \text{ a left } R\text{-module}\} \), where \( \operatorname{pd}_R(M) \) is the length of the shortest projective resolution of \( M \)<sup>[1](https://encyclopediaofmath.org/wiki/Dimension)</sup> |
| Left and right | The left and right global dimensions are defined analogously; they agree for commutative rings but can differ in general<sup>[2](https://planetmath.org/globaldimension)</sup> |
| Injective form | The supremum of injective dimensions of left modules gives the same number<sup>[1](https://encyclopediaofmath.org/wiki/Dimension)</sup> |
| Dimension 0 | \( \operatorname{gl\,dim} R = 0 \) on the left if and only if \( R \) is semisimple, if and only if the right global dimension is 0<sup>[2](https://planetmath.org/globaldimension)</sup> |
| Dimension 1 | \( \operatorname{gl\,dim} R = 1 \) on one side exactly when \( R \) is hereditary on that side and not semisimple<sup>[2](https://planetmath.org/globaldimension)</sup> |
| Regular local rings | A Noetherian local ring is regular if and only if it has finite global dimension, in which case the global dimension equals \( \dim R \)<sup>[3](https://stacks.math.columbia.edu/tag/065U)</sup> |
| Polynomial rings | Hilbert's syzygy theorem: \( \operatorname{gl\,dim} F[x_1, \ldots, x_n] = n \) for a field \( F \)<sup>[4](https://andreabianchi.sites.ku.dk/files/2023/03/HomAlg2021.pdf)</sup> |

## Definition and basic properties

A projective resolution of a left \( R \)-module \( M \) is an exact sequence

\[ \cdots \to P_n \to \cdots \to P_0 \to M \to 0 \]

in which each \( P_i \) is projective. Every module has one, and the projective dimension \( \operatorname{pd}_R(M) \) is the least \( n \) for which a resolution of length \( n \) exists; \( \operatorname{pd}_R(M) = 0 \) exactly when \( M \) is projective, and \( \operatorname{pd}_R(M) = \infty \) when no finite length suffices.<sup>[1](https://encyclopediaofmath.org/wiki/Dimension)</sup> The global dimension is then

\[ \operatorname{gl\,dim} R = \sup \{ \operatorname{pd}_R(M) : M \text{ an } R\text{-module} \}. \]

The same number results whether one takes the supremum over left or over right modules.<sup>[1](https://encyclopediaofmath.org/wiki/Dimension)</sup> For a commutative ring the two sides coincide outright, and one writes simply \( \operatorname{Gd}(R) \).<sup>[2](https://planetmath.org/globaldimension)</sup> For noncommutative rings the left and right global dimensions can differ, although vanishing on one side is equivalent to vanishing on the other.<sup>[2](https://planetmath.org/globaldimension)</sup> If \( R \) is both left and right Noetherian, the left and right global dimensions are equal.<sup>[1](https://encyclopediaofmath.org/wiki/Dimension)</sup>

**Equivalent characterizations.** The global dimension can also be computed from injective resolutions: the supremum of the [injective dimensions](https://www.edgepedia.org/injective-dimension) of all left modules equals \( \operatorname{gl\,dim} R \).<sup>[1](https://encyclopediaofmath.org/wiki/Dimension)</sup> For a ring \( R \), \( \operatorname{gl\,dim} R \le n \) is equivalent to each of the following: every finitely generated module has projective dimension at most \( n \); every cyclic module \( R/I \) has projective dimension at most \( n \); every module has injective dimension at most \( n \); and \( \operatorname{Ext}^q_R(M, N) = 0 \) for all \( q > n \) and all modules \( M \) and \( N \).<sup>[5](https://www.math.uchicago.edu/~may/MISC/RegularLocal.pdf)</sup> For a commutative Noetherian local ring, the residue field alone detects the invariant: \( \operatorname{gl\,dim}(R) = \operatorname{pd}_R(k) \), where \( k = R/\mathfrak{m} \).<sup>[5](https://www.math.uchicago.edu/~may/MISC/RegularLocal.pdf)</sup><sup> • </sup><sup>[3](https://stacks.math.columbia.edu/tag/065U)</sup>

## Weak global dimension

Replacing projective modules by flat modules gives a related invariant. The [flat dimension](https://www.edgepedia.org/flat-dimension) \( \operatorname{fd}(M) \) of a module is the length of its shortest resolution by flat modules, and the left weak dimension \( \operatorname{l.wdim}(R) \) is the supremum of \( \operatorname{fd}(M) \) over left modules, with the right weak dimension defined similarly.<sup>[1](https://encyclopediaofmath.org/wiki/Dimension)</sup> Unlike the global dimension, the weak global dimension is left-right symmetric: a ring has finite weak global dimension on the left if and only if it has the same on the right.<sup>[6](https://planetmath.org/WeakGlobalDimension)</sup><sup> • </sup><sup>[7](https://www.math.ttu.edu/~lchriste/download/ggdim.pdf)</sup> A ring is von Neumann regular if and only if its weak global dimension is 0.<sup>[8](https://mathoverflow.net/questions/495331/is-there-a-von-neumann-regular-ring-with-arbitrary-left-and-right-global-dimensi)</sup> The weak dimension can be computed from cyclic modules alone: \( \operatorname{fdim}(R) = \sup\{\operatorname{fd}(R/J)\} \) taken over left ideals \( J \), and the same supremum over right ideals gives the same value.<sup>[4](https://andreabianchi.sites.ku.dk/files/2023/03/HomAlg2021.pdf)</sup>

## Regular rings and the Auslander–Buchsbaum–Serre theorem

The global dimension became central to commutative algebra through its connection with regularity. In 1956, [Maurice Auslander](https://www.edgepedia.org/maurice-auslander) and [David Buchsbaum](https://www.edgepedia.org/david-buchsbaum), and independently [Jean-Pierre Serre](https://www.edgepedia.org/jean-pierre-serre), used homological methods to characterize regular local rings as exactly those Noetherian local rings of finite global dimension.<sup>[9](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup> In this case the global dimension is not merely finite but equals the Krull dimension of the ring, \( \dim R = \dim_k(\mathfrak{m}/\mathfrak{m}^2) \), the minimal number of generators of the maximal ideal.<sup>[3](https://stacks.math.columbia.edu/tag/065U)</sup> More precisely, for a Noetherian local ring \( (R, \mathfrak{m}, \kappa) \), the following are equivalent: the residue field \( \kappa \) has finite projective dimension, \( R \) has finite global dimension, and \( R \) is a regular local ring.<sup>[3](https://stacks.math.columbia.edu/tag/065U)</sup>

The result globalizes through localization. For a [Noetherian ring](https://www.edgepedia.org/noetherian-ring) \( R \) and an integer \( n \ge 0 \), \( R \) has global dimension at most \( n \) if and only if every localization \( R_{\mathfrak{m}} \) at a maximal ideal does.<sup>[3](https://stacks.math.columbia.edu/tag/065U)</sup> A Noetherian ring is called regular when all its localizations at primes are regular local rings, and \( R \) has finite global dimension \( n \) exactly when \( R \) is a regular ring of dimension \( n \).<sup>[3](https://stacks.math.columbia.edu/tag/065U)</sup> Regularity and finite global dimension nevertheless part company for infinite-dimensional rings: there exist regular Noetherian rings of infinite dimension, and these do not have finite global dimension.<sup>[3](https://stacks.math.columbia.edu/tag/065U)</sup>

The same circle of ideas produced the Auslander–Buchsbaum formula, an identity relating the projective dimension of a module of finite projective dimension over a commutative Noetherian local ring to its depth, and the 1959 theorem of Auslander and Buchsbaum that every regular local ring is a unique factorization domain.<sup>[10](https://ncatlab.org/nlab/show/projective%2Bdimension)</sup><sup> • </sup><sup>[9](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup>

## Examples and computations

**Dimension zero and one.** The semisimple Artinian rings are exactly the rings of global dimension zero.<sup>[1](https://encyclopediaofmath.org/wiki/Dimension)</sup> Dedekind rings are examples of rings of global dimension one.<sup>[5](https://www.math.uchicago.edu/~may/MISC/RegularLocal.pdf)</sup> On the weak side, \( \operatorname{r.Gd}(R) = 1 \) holds exactly when \( R \) is right hereditary but not semisimple.<sup>[2](https://planetmath.org/globaldimension)</sup>

**Polynomial rings.** Hilbert's syzygy theorem, proved by Hilbert in 1890, states that every finitely generated module over the polynomial ring \( F[x_1, \ldots, x_n] \) in \( n \) variables over a field \( F \) has a free resolution of length at most \( n \), and the bound is attained: \( \operatorname{gl\,dim} F[x_1, \ldots, x_n] = n \).<sup>[4](https://andreabianchi.sites.ku.dk/files/2023/03/HomAlg2021.pdf)</sup> Hilbert's original statement was that every ideal in such a ring admits a free resolution of length at most \( n - 1 \).<sup>[4](https://andreabianchi.sites.ku.dk/files/2023/03/HomAlg2021.pdf)</sup> The theorem extends to other settings: for a polynomial ring \( R[M(E,I)] \) in partially commuting variables whose independence graph has a particular four-vertex shape, \( \operatorname{gl\,dim} R[M(E,I)] = 2 + \operatorname{gl\,dim} R \).<sup>[11](https://arxiv.org/abs/0809.4518)</sup>

**Localization behavior.** If \( R \) has finite global dimension at most \( n \), then every localization \( S^{-1}R \) at a multiplicative subset \( S \) has global dimension at most \( n \).<sup>[12](https://stacks.math.columbia.edu/tag/00O2)</sup> For a commutative Noetherian local ring, the global dimension equals the projective dimension of the residue field, which is also its injective dimension.<sup>[13](https://mathoverflow.net/questions/59981/global-dimension-and-localization)</sup>

**Related invariants.** For an algebra \( R \) over a commutative ring \( K \), the projective dimension of \( R \) as a bimodule, that is, of \( R \otimes_K R^{\mathrm{op}} \), is called the bidimension \( \operatorname{bid} R \). For a group \( G \) and commutative ring \( K \), the cohomological dimension \( \operatorname{cd}(G) \) is the projective dimension of \( K \) over the group ring \( KG \) with trivial \( G \)-action.<sup>[14](https://encyclopediaofmath.org/wiki/Homological_dimension)</sup>

## History

The invariant was introduced by Maurice Auslander, a mathematician then at the University of Michigan and Princeton University, in the 1955 Nagoya Mathematical Journal paper "On the dimension of modules and algebras. III. Global dimension".<sup>[15](https://bishtref.com/articles/10.1090/s0002-9947-1957-0086822-7)</sup> Auslander and Buchsbaum pursued homological dimension in Noetherian rings in a 1956 [Proceedings of the National Academy of Sciences](https://www.edgepedia.org/proceedings-of-the-national-academy-of-sciences) note and in "Homological dimension in local rings" (Transactions of the American Mathematical Society, 1957).<sup>[16](https://pmc.ncbi.nlm.nih.gov/articles/PMC534227/)</sup><sup> • </sup><sup>[15](https://bishtref.com/articles/10.1090/s0002-9947-1957-0086822-7)</sup> The left and right global dimensions of a ring were early targets of this theory, and the Auslander–Buchsbaum–Serre characterization of regular local rings by finite global dimension became one of its principal applications.<sup>[17](https://sites.math.rutgers.edu/~weibel/HA-history.pdf)</sup><sup> • </sup><sup>[9](https://www.sciencedirect.com/topics/mathematics/regular-sequence)</sup>

## References

1. [Dimension, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Dimension)
2. [Global dimension, PlanetMath](https://planetmath.org/globaldimension)
3. [Section 10.110: Regular rings and global dimension, The Stacks Project](https://stacks.math.columbia.edu/tag/065U)
4. [Andrea Bianchi, Lecture notes for Homological Algebra (2021)](https://andreabianchi.sites.ku.dk/files/2023/03/HomAlg2021.pdf)
5. [Peter May, Regular local rings and global dimension, University of Chicago notes](https://www.math.uchicago.edu/~may/MISC/RegularLocal.pdf)
6. [Weak global dimension, PlanetMath](https://planetmath.org/WeakGlobalDimension)
7. [L. W. Christensen, S. Estrada, P. Thompson, Gorenstein weak global dimension is symmetric](https://www.math.ttu.edu/~lchriste/download/ggdim.pdf)
8. [MathOverflow: von Neumann regular rings and global dimensions](https://mathoverflow.net/questions/495331/is-there-a-von-neumann-regular-ring-with-arbitrary-left-and-right-global-dimensi)
9. [Regular Sequence overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/regular-sequence)
10. [Projective dimension, nLab](https://ncatlab.org/nlab/show/projective%2Bdimension)
11. [A. A. Husainov, Global Dimension of Polynomial Rings in Partially Commuting Variables, arXiv:0809.4518](https://arxiv.org/abs/0809.4518)
12. [Section 10.109: Rings of finite global dimension, The Stacks Project](https://stacks.math.columbia.edu/tag/00O2)
13. [MathOverflow: Global dimension and localization](https://mathoverflow.net/questions/59981/global-dimension-and-localization)
14. [Homological dimension, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Homological_dimension)
15. [Homological dimension in local rings, Auslander & Buchsbaum, Trans. AMS 85 (1957)](https://bishtref.com/articles/10.1090/s0002-9947-1957-0086822-7)
16. [M. Auslander, D. A. Buchsbaum, Homological dimension in Noetherian rings, PNAS 42 (1956)](https://pmc.ncbi.nlm.nih.gov/articles/PMC534227/)
17. [Charles Weibel, History of Homological Algebra](https://sites.math.rutgers.edu/~weibel/HA-history.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Ring theory › Commutative algebra › Regular rings and homological properties*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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