In this note, we give self-contained proofs of the following
three theorems:
Theorem A (Auslander-Buchsbaum formula) The depth of a
Noetherian local ring A is equal to the sum of the
depth and the projective dimension of a nitely
generated A-module M with nite projective dimension.
Theorem B (Serre theorem) A Noetherian local ring is regular
if and only if its global dimension is nite.
Theorem C A regular local ring is a unique factorization domain.
