Regular Local Ring - Characterizations

Characterizations

There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if is a Noetherian local ring with maximal ideal, then the following are equivalent definitions

• Let where is chosen as small as possible. Then is regular if

,

where the dimension is the Krull dimension. The minimal set of generators of are then called a regular system of parameters.

• Let be the residue field of . Then is regular if

,

where the second dimension is the Krull dimension.

• Let be the global dimension of (i.e., the supremum of the projective dimensions of all -modules.) Then is regular if

,

in which case, .