Global Dimension

:gl dim(

A

),
is the Supremum of the set of Projective Dimensions of all

A

- Modules . By Homological Algebra , this is equal to:

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

.

REFERENCES

Hideyuki Matsumura, ''Commutative Ring Theory'', Cambridge studies in advanced mathematics 8.

	CATEGORIES ABOUT GLOBAL DIMENSION

	ring theory

	module theory

	homological algebra

	dimension

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