Global dimension

Remarks

By a theorem of Auslander, the supremum can be taken over the cyclic modules $R/I$ only, with $I$ an ideal. For 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. Example: $\operatorname{gl.dim} \Z = 1$; 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. Example: 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. Non-example of finiteness: $\Z/4\Z$ has infinite global dimension, because the projective dimension of $\Z/2\Z$ over it is infinite. For a Noetherian local ring $(R, \mathfrak{m}, k)$, the global dimension equals $\operatorname{pd}_R k$.