# 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].