 A local Noetherian ring is regular if and only if its global dimension is finite, say "n", which means that any finitely generated "R"-module has a resolution by projective modules of length at most "n".The proof of this and other related statements relies on the usage of homological methods, such as the
Ext functor.This functor is the derived functor of the functor
The latter functor is exact if "M" is projective, but not otherwise