PROPOSITION 10.1Let
be a local ring with the residue field
. Let
be
-modules.
We assume
is a finite module. Then an
-module homomorphism
is surjective if and only if
is surjective.

DEFINITION 10.2
Let

be a resolution of
. We say that it is minimal free resolution if
the following conditions are satisfied.

each
is a finite free
-module.

.

is an isomorphism.

LEMMA 10.3A finite module
over a Noetherian local ring has a minimal free resolution.

DEFINITION 10.4
Let
be an
-module over a ring
. We define the projective dimension of
(
)
to be the minimal length of the projective resolution of
.

PROPOSITION 10.5Let
be a local ring with the residue field
. Then:

If we have a minimal resolution
of an
-module
, then we have
.

.

DEFINITION 10.6
We define the global dimension of a ring
by

LEMMA 10.7Let
be a local ring. Let
be its residue field.
Then we have

THEOREM 10.8 (Serre)
Let
be a Noetherian local ring. Then:

is regular

THEOREM 10.9 (Serre)
A localization of a regular local ring at a prime ideal is also a regular local ring.

DEFINITION 10.10
A Noetherian ring is said to be regular if its localilzation at every prime is a regular local ring.