Buchsbaum ring
The notion of a Buchsbaum ring (and module) is a generalization of that of a Cohen–Macaulay ring (respectively, module). Let denote a Noetherian local ring (cf. also Noetherian ring) with maximal ideal
and
. Let
be a finitely-generated
-module with
. Then
is called a Buchsbaum module if the difference
![]() |
is independent of the choice of a parameter ideal of
, where
is a system of parameters of
and
(respectively,
) denotes the length of the
-module
(respectively, the multiplicity of
with respect to
). When this is the case, the difference
![]() |
is called the Buchsbaum invariant of . The
-module
is a Cohen–Macaulay module if and only if
for some (and hence for any) parameter ideal
of
, so that
is a Cohen–Macaulay
-module if and only if
is a Buchsbaum
-module with
. The ring
is said to be a Buchsbaum ring if
is a Buchsbaum module over itself. If
is a Buchsbaum ring, then
is a Cohen–Macaulay ring with
for every
.
A typical example of Buchsbaum rings is as follows. Let
![]() |
where , with
, denotes the formal power series ring in
variables over a field
. Then
is a Buchsbaum ring with
and
.
A, not necessarily local, Noetherian ring is said to be a Buchsbaum ring if the local rings
are Buchsbaum for all
.
The theory of Buchsbaum rings and modules dates back to a question raised in 1965 by D.A. Buchsbaum [a3]. He asked whether the difference , with
a parameter ideal, is an invariant for any Noetherian local ring
. This is, however, not the case and a counterexample was given in [a28]. Thereafter, in 1973 J. Stückrad and W. Vogel published the classic paper [a29], from which the history of Buchsbaum rings and modules started. In [a29] they gave a characterization of Buchsbaum rings in terms of the following property of systems of parameters: A
-dimensional Noetherian local ring
with maximal ideal
is Buchsbaum if and only if every system
of parameters for
forms a weak
-sequence, that is, the equality
![]() |
holds for all . Therefore, systems
of parameters in a Buchsbaum local ring need not be regular sequences, but the differences
![]() |
are very small and only finite-dimensional vector spaces over the residue class field of
. Weak sequences are closely related to
-sequences introduced by C. Huneke [a21]. Actually,
is a Buchsbaum ring if and only if every system
of parameters for
forms a
-sequence, that is, the equality
![]() |
holds for all .
One of the fundamental results on Buchsbaum rings and modules is the surjectivity criterion. Let
![]() |
denote the th local cohomology of
with respect to the maximal ideal
. If
is a Buchsbaum
-module, then
for all
and the equality
![]() |
holds, where .
Unfortunately, the vanishing does not characterize Buchsbaum modules. Modules with
for all
are called quasi-Buchsbaum and constitute a class which is strictly larger than that of Buchsbaum modules. However, if the canonical homomorphism
![]() |
is surjective for all , then
is a Buchsbaum
-module. The converse is also true if the base ring
is regular (cf. also Regular ring (in commutative algebra)).
After the establishment of the surjectivity criterion, by Stückrad and Vogel [a30] in 1978, the development of the theory became rather rapid. The ubiquity of Buchsbaum normal local rings was established by S. Goto [a6] as an application of the Evans–Griffith construction [a5]. Namely, let and
be integers. Then there exists a Buchsbaum local ring
with
and
for
. If
(respectively,
and
), one may choose the ring
so that
is an integral domain (respectively, a normal ring). See [a1] for progress in the research about the ubiquity of Buchsbaum homogeneous integral domains. Besides, Buchsbaum local rings of multiplicity
have been classified [a8]. Also, certain famous isolated singularities are Buchsbaum (cf. [a23]).
The theory of Buchsbaum rings and modules is closely related to that of Cohen–Macaulayness in blowing-ups. Let be an ideal of positive height in a Noetherian local ring
. Let
and call it the Rees algebra of
. Then the canonical morphism
is the blowing-up of
with centre
(cf. also Blow-up algebra). If the ring
is Cohen–Macaulay, then the scheme
naturally is locally Cohen–Macaulay. The problem when the Rees algebra
is Cohen–Macaulay has been intensively studied from the 1980s onwards ([a18], [a38], [a16], [a39], [a17]).
The ring is Cohen–Macaulay if the ideal
is generated by a regular sequence and if the base ring
is Cohen–Macaulay [a2]. However, the converse is not true even for parameter ideals
. Actually,
is a Buchsbaum ring if and only if the Rees algebra
is a Cohen–Macaulay ring for every parameter ideal
in
, provided that
is an integral domain with
. This insightful result of Y. Shimoda [a35] in 1979 opened the door towards a further development of the theory. Firstly, Goto and Shimoda [a19] showed that a Noetherian local ring
is a Buchsbaum ring with
(
) if and only if the Rees algebra
is a Cohen–Macaulay ring for every parameter ideal
in
. When this is the case, the Rees algebras
are also Cohen–Macaulay for all
. In 1981, Buchsbaum rings were characterized in terms of the blowing-ups of parameter ideals. Let
be a Noetherian local ring with maximal ideal
and
. Then
is a Buchsbaum ring if and only if the scheme
is locally Cohen–Macaulay for every parameter ideal
in
[a7]. Subsequently, Goto [a10] proved that the associated graded rings
of parameter ideals
in a Buchsbaum local ring are always Buchsbaum. In addition, Stückrad showed that
is a Buchsbaum ring for every parameter ideal
in a Buchsbaum local ring [a36]. The systems of parameters in Buchsbaum local rings behave very well and enjoy the monomial property [a10].
Buchsbaum rings are yet (2000) the only non-trivial case for which the monomial conjecture, raised by M. Hochster, has been solved affirmatively (except for the equi-characteristic case). See [a31] for these results, together with geometric applications and concrete examples. See [a31] for researches on the Buchsbaum property in affine semi-group rings and Stanley–Reisner rings of simplicial complexes.
Let be a Buchsbaum module over a Noetherian local ring
. Then
is said to be maximal if
. Noetherian local rings possessing only finitely many isomorphism classes of indecomposable maximal Buchsbaum modules are said to have finite Buchsbaum-representation type. Buchsbaum representation theory was studied by Goto and K. Nishida [a15], [a11], [a13], and the Cohen–Macaulay local rings
of finite Buchsbaum-representation type have been classified under certain mild conditions. If
, then
must be regular [a15]. The situation is a little more complicated if
[a13]. In [a11] (not necessarily Cohen–Macaulay) surface singularities of finite Buchsbaum-representation type are classified.
Suppose that is a regular local ring with
and let
be a maximal Buchsbaum
-module. Then
is a free
-module for all
, so that the
-module
defines a vector bundle on the punctured spectrum
of
. Thanks to the surjectivity criterion, one can prove the structure theorem of maximal Buchsbaum modules over regular local rings: Every maximal Buchsbaum
-module
has the form
![]() |
where denotes the
th syzygy module of the residue class field
of
,
(
), and
, if
is a regular local ring ([a4], [a12]). This result has been generalized by Y. Yoshino [a40] and T. Kawasaki [a24]. They showed a similar decomposition theorem of a special kind of maximal Buchsbaum modules over Gorenstein local rings; see [a32] for the characterization of Buchsbaum rings and modules in terms of dualizing complexes. (It should be noted here that the main result in [a32] contains a serious mistake, which has been repaired in [a40].)
A local ring satisfying the condition that all the local cohomology modules
(
) are finitely generated is said to be an FLC ring (or a generalized Cohen–Macaulay ring). The class of FLC rings includes Buchsbaum rings as typical examples. In fact, a Noetherian local ring
is FLC if and only if it contains at least one system
(
) of parameters such that the sequence
forms a
-sequence in any order for all integers
. Such a sequence is called an unconditioned strong
-sequence (for short, USD-sequence or
-sequence); they have been intensively studied [a27], [a37], [a20]. Recently (1999), Kawasaki [a25] used the results in [a20] to establish the arithmetic Cohen–Macaulayfications of Noetherian local rings. Namely, every unmixed local ring
contains an ideal
of positive height with the Cohen–Macaulay Rees algebra
, provided
and all the formal fibres of
are Cohen–Macaulay. Hence, the Sharp conjecture [a34] concerning the existence of dualizing complexes is solved affirmatively.
Let be a Noetherian graded ring with
a field and let
. Then
is a Buchsbaum ring if and only if the local ring
is Buchsbaum. When this is the case, the local cohomology modules
(
) are finite-dimensional vector spaces over the field
. The vanishing of certain homogeneous components
of
may affect the Buchsbaumness in graded algebras
. For example, if there exist integers
(
) such that
for all
and if
![]() |
for all and
, then
is a Buchsbaum ring [a9]. Therefore
is a Buchsbaum ring if
for all
[a33]. Hence the scheme
is arithmetically Buchsbaum if
is locally Cohen–Macaulay, provided that
and
is equi-dimensional. See [a22] for the bounds of Castelnuovo–Mumford regularities of Buchsbaum schemes
.
Researches of the Buchsbaumness in Rees algebras recently (1999) started again, although the progress remains tardy (possibly because of the lack of characterizations of Trung–Ikeda type [a38] for Buchsbaumness). In [a14] the Buchsbaumness in Rees algebras of certain
-primary ideals
in Cohen–Macaulay local rings is closely studied in connection with the Buchsbaumness in the associated graded rings
and that of the extended Rees algebras
. In [a26], [a41], [a42], Buchsbaumness in graded rings associated to certain
-primary ideals in Buchsbaum local rings is explored. Especially, the Rees algebra
of the maximal ideal
in a Buchsbaum local ring
of maximal embedding dimension (that is, a Buchsbaum local ring
for which the equality
holds) is again a Buchsbaum ring [a42].
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Buchsbaum ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Buchsbaum_ring&oldid=24390