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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]].\nThe *global dimension* of $R$ is\n$$\\operatorname{gl.dim} R = \\sup \\set{\\operatorname{pd}_R M : M \\text{ a nonzero } R\\text{-module}},$$\nwhere $\\operatorname{pd}_R M$ is the [[projective dimension|MR9QCH1PNA]] of the [[$R$-module|0NZ8JA25WK]] $M$.\nIt is an integer $\\ge 0$ or $\\infty$.\n\n## Remarks\n\nBy a theorem of Auslander, the supremum can be taken over the cyclic modules $R/I$ only, with $I$ an ideal.\nFor a ring that is not commutative, the same definition with left modules or right modules gives the left and right global dimensions, which can differ.\nExample: $\\operatorname{gl.dim} \\Z = 1$; every subgroup of a free abelian group is free, so every module has a free resolution of length at most $1$, and $\\Z/2\\Z$ is not projective.\nExample: a ring has global dimension $0$ if and only if every module is projective; for a field this holds since every vector space is free.\nNon-example of finiteness: $\\Z/4\\Z$ has infinite global dimension, because the projective dimension of $\\Z/2\\Z$ over it is infinite.\nFor a Noetherian local ring $(R, \\mathfrak{m}, k)$, the global dimension equals $\\operatorname{pd}_R k$.\n",
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