Global Dimension¶
A ring's global dimension is the supremum of projective-resolution lengths across all modules on a specified side.
Core Idea¶
The global dimension of a ring is the supremum of the shortest projective-resolution lengths of all its modules. For a noncommutative ring, specify left or right modules. If there is no finite uniform bound, the value is infinity.[^ref-9def0bf3c568] A field has dimension zero because every module over it is free.
Scope of Application¶
In homological algebra, the invariant measures ring-wide projective resolution complexity. For \(R=k[x,y]\), the residue module \(k=R/(x,y)\) has the exact free Koszul resolution \(0\to R\xrightarrow{f\mapsto(-yf,xf)}R^2\xrightarrow{(a,b)\mapsto xa+yb}R\to k\to0\), so its projective dimension is at most 2. Localizing at \((x,y)\), the Stacks embedding-dimension lower bound gives at least 2; hence this module attains 2.[^ref-4319bf5a46ed] Stacks' polynomial-ring theorem supplies the upper bound for every module, so global dimension is 2.[^ref-87142265597b] For a commutative Noetherian local ring, finite global dimension is equivalent to regularity and then equals Krull dimension; that is a conditional theorem, not the definition.[^ref-4319bf5a46ed]
Clarity¶
First minimize a resolution length for each module; then take a supremum across every module on the selected side. One module's short resolution does not give the ring's global bound. Weak dimension instead uses flat resolutions.
Manages Complexity¶
One number summarizes whether projective resolutions have a common finite upper length. This exact invariant has no universal intrinsic opposed-cost tension: one module versus all modules, left versus right, and projective versus flat are quantifier or carrier distinctions rather than design choices. The summary is useful only if “all modules” and the projective carrier are preserved.
Abstract Reasoning¶
To prove an upper bound \(n\), bound projective dimensions of every module. To prove a lower bound, find a module requiring at least \(n\) steps. Use local regularity or geometric dimension comparisons only under their stated algebraic hypotheses.[ref-56e14fb20c84][ref-4319bf5a46ed]
Knowledge Transfer¶
The portable skeleton is minimum length per object followed by the categorywide supremum. Its exact projective modules, resolutions and ring side make it a domain-specific homological invariant rather than a generic prime for worst-case difficulty. Flat-resolution Weak Dimension is a neighbor, not a strict parent; no immediate current parent is asserted. Commutative regular-local comparisons do not transfer without their hypotheses.
[^ref-56e14fb20c84]: The Stacks Project, Algebra §10.109, “Rings of finite global dimension”, especially Definitions 10.109.2 and 10.109.10 and the finite-bound criteria.
[^ref-4319bf5a46ed]: The Stacks Project, Algebra §10.110, “Regular rings and global dimension”, especially Proposition 10.110.5 and the warning before Lemma 10.110.8.
[^ref-87142265597b]: The Stacks Project, Algebra §10.114, “Dimension of finite type algebras over fields”, especially Proposition 10.114.2 on polynomial-ring global dimension.
[^ref-9def0bf3c568]: Auslander, “On the Dimension of Modules and Algebras (III): Global Dimension,” Nagoya Mathematical Journal 9 (1955): 67–77, p. 67 for the unital-ring left/right definitions.
Neighborhood in Abstraction Space¶
Global Dimension sits in a sparse region of the domain-specific corpus (89th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Cohomological dimension — 0.83
- Projective variety — 0.80
- Weak dimension — 0.80
- Perfect ring — 0.79
- Finite morphism — 0.79
Computed from structural-signature embeddings · 2026-10-08