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
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].
|[a1]||M. Amasaki, "Existence of homogeneous prime ideals fitting into long Bourbaki sequences" , Proc. 21st Symp. Commutative Algebra in Tokyo, Japan, November 23-26, 1999 (1999) pp. 104–111 MR1610925|
|[a2]||J. Barshay, "Graded algebras of powers of ideals generated by A-sequences" J. Algebra , 25 (1973) pp. 90–99 MR0332748 Zbl 0256.13017|
|[a3]||D.A. Buchsbaum, "Complexes in local ring theory" , Some Aspects of Ring Theory , C.I.M.E. Roma (1965) pp. 223–228 Zbl 0178.37201|
|[a4]||G. Eisenbud, S. Goto, "Linear free resolutions and minimal multiplicity" J. Algebra , 88 (1984) pp. 89–133 MR0741934 Zbl 0531.13015|
|[a5]||E.G. Evans Jr., P.A. Griffith, "Local cohomology modules for normal domains" J. London Math. Soc. , 19 (1979) pp. 277–284 MR0533326 Zbl 0407.13019|
|[a6]||S. Goto, "On Buchsbaum rings" J. Algebra , 67 (1980) pp. 272–279 MR0602063 Zbl 0473.13010 Zbl 0473.13009 Zbl 0413.13012|
|[a7]||S. Goto, "Blowing-up of Buchsbaum rings" , Commutative Algebra , Lecture Notes , 72 , London Math. Soc. (1981) pp. 140–162 MR0693633 Zbl 0519.13021|
|[a8]||S. Goto, "Buchsbaum rings with multiplicity 2" J. Algebra , 74 (1982) pp. 494–508 MR0647250 Zbl 0479.13007|
|[a9]||S. Goto, "Buchsbaum rings of maximal embedding dimension" J. Algebra , 76 (1982) pp. 383–399 MR0661862 Zbl 0482.13012|
|[a10]||S. Goto, "On the associated graded rings of parameter ideals in Buchsbaum rings" J. Algebra , 85 (1983) pp. 490–534 MR0725097 Zbl 0529.13010|
|[a11]||S. Goto, "Surface singularities of finite Buchsbaum-representation type" , Commutative Algebra: Proc. Microprogram June 15–July 2 , Springer (1987) pp. 247–263 MR1015521 Zbl 0741.13015|
|[a12]||S. Goto, "Maximal Buchsbaum modules over regular local rings and a structure theorem for generalized Cohen–Macaulay modules" M. Nagata (ed.) H. Matsumura (ed.) , Commutative Algebra and Combinatorics , Adv. Stud. Pure Math. , 11 , Kinokuniya (1987) pp. 39–46 MR0951196 Zbl 0649.13009|
|[a13]||S. Goto, "Curve singularities of finite Buchsbaum-representation type" J. Algebra , 163 (1994) pp. 447–480 MR1262714 Zbl 0807.13007|
|[a14]||S. Goto, "Buchsbaumness in Rees algebras associated to ideals of minimal multiplicity" J. Algebra , 213 (1999) pp. 604–661 MR1673472 Zbl 0942.13003|
|[a15]||S. Goto, K. Nishida, "Rings with only finitely many isomorphism classes of indecomposable maximal Buchsbaum modules" J. Math. Soc. Japan , 40 (1988) pp. 501–518 MR0945349 Zbl 0657.13022|
|[a16]||S. Goto, K. Nishida, "The Cohen–Macaulay and Gorenstein Rees algebras associated to filtrations" , Memoirs , 526 , Amer. Math. Soc. (1994) MR1287443 Zbl 0812.13016|
|[a17]||S. Goto, Y. Nakamura, K. Nishida, "Cohen–Macaulay graded rings associated ideals" Amer. J. Math. , 118 (1996) pp. 1197–1213 Zbl 0878.13002|
|[a18]||S. Goto, Y. Shimoda, "On the Rees algebras of Cohen–Macaulay local rings" R.N. Draper (ed.) , Commutative Algebra, Analytic Methods , Lecture Notes in Pure Applied Math. , 68 , M. Dekker (1982) pp. 201–231 MR0655805 Zbl 0482.13011|
|[a19]||S. Goto, Y. Shimoda, "On Rees algebras over Buchsbaum rings" J. Math. Kyoto Univ. , 20 (1980) pp. 691–708 MR0592354 Zbl 0473.13010|
|[a20]||S. Goto, K. Yamagishi, "The theory of unconditioned strong -sequences and modules of finite local cohomology" Preprint (1978)|
|[a21]||C. Huneke, "The theory of d-sequences and powers of ideals" Adv. Math. , 46 (1982) pp. 249–279 MR0683201 Zbl 0505.13004|
|[a22]||L.T. Hoa, C. Miyazaki, "Bounds on Castelnuovo–Mumford regularity for generalized Cohen–Macaulay graded rings" Math. Ann. , 301 (1995) pp. 587–598 MR1324528 Zbl 0834.13016|
|[a23]||M.-N. Ishida, "Tsuchihashi's cusp singularities are Buchsbaum singularities" Tôhoku Math. J. , 36 (1984) pp. 191–201 MR742594|
|[a24]||T. Kawasaki, "Local cohomology modules of indecomposable surjective–Buchsbaum modules over Gorenstein local rings" J. Math. Soc. Japan , 48 (1996) pp. 551–566 MR1389995 Zbl 0866.13007|
|[a25]||T. Kawasaki, "Arithmetic Cohen–Macaulayfications of local rings" , Proc. 21st Symp. Commutative Algebra in Tokyo, Japan, November 23-26, 1999 (1999) pp. 88–92|
|[a26]||Y. Nakamura, "On the Buchsbaum property of associated graded rings" J. Algebra , 209 (1998) pp. 345–366 MR1652142 Zbl 0942.13002|
|[a27]||P. Schenzel, N.V. Trung, N.T. Cuong, "Verallgemeinerte Cohen–Macaulay-Moduln" Math. Nachr. , 85 (1978) pp. 57–73 MR0517641 Zbl 0398.13014|
|[a28]||J. Stückrad, W. Vogel, "Ein Korrekturglied in der Multiplizitätstheorie von D.G. Northcott und Anwendungen" Monatsh. Math. , 76 (1972) pp. 264–271 Zbl 0248.13026|
|[a29]||J. Stückrad, W. Vogel, "Eine Verallgemeinerung der Cohen–Macaulay-Ringe und Anwendungen auf ein Problem der Multiplitätstheorie" J. Math. Kyoto Univ. , 13 (1973) pp. 513–528|
|[a30]||J. Stückrad, W. Vogel, "Toward a theory of Buchsbaum singularities" Amer. J. Math. , 100 (1978) pp. 727–746 MR0509072 Zbl 0429.14001|
|[a31]||J. Stückrad, W. Vogel, "Buchsbaum rings and applications" , Springer (1986) MR0881220 MR0873945 Zbl 0606.13018 Zbl 0606.13017|
|[a32]||P. Schenzel, "Dualisierende Komplexe in der lokalen Algebra und Buchsbaum–Ringe" , Lecture Notes in Mathematics , 907 , Springer (1982) MR0654151 Zbl 0484.13016|
|[a33]||P. Schenzel, "On Veronesean embeddings and projections of Veronesean varieties" Archiv Math. , 30 (1978) pp. 391–397 MR0485849 Zbl 0417.14040|
|[a34]||R.Y. Sharp, "Necessary conditions for the existence of dualizing complexes in commutative algebra" , Lecture Notes in Mathematics , 740 , Springer (1979) pp. 213–229 MR0563505 Zbl 0421.13003|
|[a35]||Y. Shimoda, "A note on Rees algebras of two-dimensional local domains" J. Math. Kyoto Univ. , 19 (1979) pp. 327–333 MR0545713 Zbl 0447.13010|
|[a36]||J. Stückrad, "On the Buchsbaum property of Rees and form modules" Beitr. Algebra Geom. , 19 (1985) pp. 83–103 MR0785248 Zbl 0567.13008|
|[a37]||N.V. Trung, "Toward a theory of generalized Cohen–Macaulay modules" Nagoya Math. J. , 102 (1986) pp. 1–49 Zbl 0649.13008 Zbl 0637.13013|
|[a38]||N.V. Trung, S. Ikeda, "When is the Rees algebra Cohen–Macaulay?" Commun. Algebra , 17 (1989) pp. 2893–2922 Zbl 0696.13015|
|[a39]||W. Vasconcelos, "Arithmetic of blowup algebras" , London Math. Soc. Lecture Notes , 195 , Cambridge Univ. Press (1994) MR1275840 Zbl 0813.13008|
|[a40]||Y. Yoshino, "Maximal Buchsbaum modules of finite projective dimension" J. Algebra , 159 (1993) pp. 240–264 MR1231212 Zbl 0791.13009|
|[a41]||K. Yamagishi, "The associated graded modules of Buchsbaum modules with respect to -primary ideals in equi-I-invariant case" J. Algebra , 225 (2000) pp. 1–27 MR1743648|
|[a42]||K. Yamagishi, "Buchsbaumness in Rees algebras associated to -primary ideals of minimal multiplicity in Buchsbaum local rings" , Proc. 21st Symp. Commutative Algebra in Tokyo, Japan, November 23-26, 1999 (1999) pp. 39–45|
Buchsbaum ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Buchsbaum_ring&oldid=24390