commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

, the Auslander–Buchsbaum formula, introduced by , states that if ''R'' is a commutative

Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...

local ring In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...

and ''M'' is a non-zero finitely generated ''R''-module of finite

projective dimension In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various equivalent characterizatio ...

, then: :

\mathrm_R(M) + \mathrm(M) = \mathrm(R).

Here pd stands for the projective dimension of a module, and depth for the depth of a module.

Applications

The Auslander–Buchsbaum theorem implies that a Noetherian local ring is regular if, and only if, it has finite

global dimension In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring ''A'' denoted gl dim ''A'', is a non-negative integer or infinity which is a homological invaria ...

. In turn this implies that the localization of a regular local ring is regular. If ''A'' is a local finitely generated ''R''-algebra (over a regular local ring ''R''), then the Auslander–Buchsbaum formula implies that ''A'' is Cohen–Macaulay if, and only if, pd_''R''''A'' = codim_''R''''A''.

References

* *Chapter 19 of Commutative algebra Theorems in ring theory {{commutative-algebra-stub