Global Dimension - Alternative Characterizations

Alternative Characterizations

The right global dimension of a ring A can be alternatively defined as:

• the supremum of the set of projective dimensions of all cyclic right A-modules;
• the supremum of the set of projective dimensions of all finite right A-modules;
• the supremum of the injective dimensions of all right A-modules;
• when A is a commutative Noetherian local ring with maximal ideal m, the projective dimension of the residue field A/m.

The left global dimension of A has analogous characterizations obtained by replacing "right" with "left" in the above list.

Serre proved that a commutative Noetherian local ring A is regular if and only if it has finite global dimension, in which case the global dimension coincides with the Krull dimension of A. This theorem opened the door to application of homological methods to commutative algebra.