The theorem that over a ring
of polynomials in several variables over a field every
finitely-generated projective module is free is known as the
Quillen–Suslin theorem.
The question was raised by
J.P. Serre
in
1955,
[a2],
and the statement is also still known as
Serre's conjecture.
For a complete and detailed discussion, cf.
[a3].
In
[a5],
the Quillen–Suslin theorem is formulated as: If
is a finitely-generated projective
-module
and
is a monic polynomial such that
is a free
-module,
then
is a free
-module.
Quillen's proof of the Quillen–Suslin theorem uses
Horrock's theorem:
Let
be a commutative local ring and
a finitely-generated projective module over
.
Then if
is a free
-module,
is a free
-module.
A second main ingredient is
Quillen's patching theorem.
Let
be a ring. An
-module
is
extended
(from
)
if there exists an
-module
such that
.
The patching theorem now says that if
is a commutative ring and
is a finitely-presented
-module,
then
is extended from
if and only if for every maximal ideal
of
the localization
is extended from
.
In this terminology one has a
generalized Quillen–Suslin theorem:
If
is a commutative regular ring of Krull dimension
2, then every finitely-generated projective module over
is extended from
.
The
Murthy–Horrock theorem
says that every finitely-generated projective module over
is free if
is a commutative regular local ring of Krull dimension 2.
The
Suslin monic polynomial theorem
played a major role in the study of cancellation theorems over
.
(Cancellation theorems
are theorems of the type: If
,
then
.
For instance, there is the
Bass cancellation theorem,
which says that if
is a commutative Noetherian ring of Krull dimension
and
are finitely-generated projective modules which are
stably isomorphic,
i.e.
for some
,
and the rank of
is
,
then
.)
The monic polynomial theorem says that if
is a commutative Noetherian ring of Krull dimension
and
is an ideal in
of height
,
then there exist new variables
in
such that
and such that
contains a polynomial which is monic as a polynomial in
.
For
a field this essentially becomes the
Noether normalization theorem.
A commutative ring
is said to be a
Hermite ring
if every finitely-generated
stably free module
(i.e.
for some
)
is free.
Serre's conjecture does not necessarily hold for
if
and
is a (non-commutation) division ring,
[a4].
The
quadratic analogue of Serre's conjecture
asks whether a finitely-generated projective module over
equipped with a quadratic, symmetric bilinear, or symplectic form
is necessarily extended from a similar object over
.
This is not always the case, cf.
[a3],
Chapt. VI, for more details.