Global dimension
From Wikipedia, the free encyclopedia
In mathematics, more specifically in abstract algebra, the global dimension of a ring A, denoted
- gl dim(A),
is the supremum of the set of projective dimensions of all A-modules. By homological algebra, this is equal to:
- the supremum of the set of projective dimensions of all finite A-modules
- the supremum of the injective dimensions of all A-modules.
- the projective dimension of the residue field A / m, when A is a commutative Noetherian local ring with maximal ideal m.
As an application to commutative algebra, 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 is precisely the Krull dimension of A.
[edit] References
- Hideyuki Matsumura, Commutative Ring Theory, Cambridge studies in advanced mathematics 8.
