Difference between revisions of "Buchsbaum ring"
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> 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 {{MR|1610925}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Barshay, "Graded algebras of powers of ideals generated by A-sequences" ''J. Algebra'' , '''25''' (1973) pp. 90–99 {{MR|0332748}} {{ZBL|0256.13017}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D.A. Buchsbaum, "Complexes in local ring theory" , ''Some Aspects of Ring Theory'' , C.I.M.E. Roma (1965) pp. 223–228 {{MR|}} {{ZBL|0178.37201}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> G. Eisenbud, S. Goto, "Linear free resolutions and minimal multiplicity" ''J. Algebra'' , '''88''' (1984) pp. 89–133 {{MR|0741934}} {{ZBL|0531.13015}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E.G. Evans Jr., P.A. Griffith, "Local cohomology modules for normal domains" ''J. London Math. Soc.'' , '''19''' (1979) pp. 277–284 {{MR|0533326}} {{ZBL|0407.13019}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> S. Goto, "On Buchsbaum rings" ''J. Algebra'' , '''67''' (1980) pp. 272–279 {{MR|0602063}} {{ZBL|0473.13010}} {{ZBL|0473.13009}} {{ZBL|0413.13012}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> S. Goto, "Blowing-up of Buchsbaum rings" , ''Commutative Algebra'' , ''Lecture Notes'' , '''72''' , London Math. Soc. (1981) pp. 140–162 {{MR|0693633}} {{ZBL|0519.13021}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> S. Goto, "Buchsbaum rings with multiplicity 2" ''J. Algebra'' , '''74''' (1982) pp. 494–508 {{MR|0647250}} {{ZBL|0479.13007}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> S. Goto, "Buchsbaum rings of maximal embedding dimension" ''J. Algebra'' , '''76''' (1982) pp. 383–399 {{MR|0661862}} {{ZBL|0482.13012}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> S. Goto, "On the associated graded rings of parameter ideals in Buchsbaum rings" ''J. Algebra'' , '''85''' (1983) pp. 490–534 {{MR|0725097}} {{ZBL|0529.13010}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> S. Goto, "Surface singularities of finite Buchsbaum-representation type" , ''Commutative Algebra: Proc. Microprogram June 15–July 2'' , Springer (1987) pp. 247–263 {{MR|1015521}} {{ZBL|0741.13015}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> 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 {{MR|0951196}} {{ZBL|0649.13009}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> S. Goto, "Curve singularities of finite Buchsbaum-representation type" ''J. Algebra'' , '''163''' (1994) pp. 447–480 {{MR|1262714}} {{ZBL|0807.13007}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> S. Goto, "Buchsbaumness in Rees algebras associated to ideals of minimal multiplicity" ''J. Algebra'' , '''213''' (1999) pp. 604–661 {{MR|1673472}} {{ZBL|0942.13003}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> 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 {{MR|0945349}} {{ZBL|0657.13022}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> S. Goto, K. Nishida, "The Cohen–Macaulay and Gorenstein Rees algebras associated to filtrations" , ''Memoirs'' , '''526''' , Amer. Math. Soc. (1994) {{MR|1287443}} {{ZBL|0812.13016}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> S. Goto, Y. Nakamura, K. Nishida, "Cohen–Macaulay graded rings associated ideals" ''Amer. J. Math.'' , '''118''' (1996) pp. 1197–1213 {{MR|}} {{ZBL|0878.13002}} </TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> 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 {{MR|0655805}} {{ZBL|0482.13011}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> S. Goto, Y. Shimoda, "On Rees algebras over Buchsbaum rings" ''J. Math. Kyoto Univ.'' , '''20''' (1980) pp. 691–708 {{MR|0592354}} {{ZBL|0473.13010}} </TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> S. Goto, K. Yamagishi, "The theory of unconditioned strong <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290227.png" />-sequences and modules of finite local cohomology" ''Preprint'' (1978)</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> C. Huneke, "The theory of d-sequences and powers of ideals" ''Adv. Math.'' , '''46''' (1982) pp. 249–279 {{MR|0683201}} {{ZBL|0505.13004}} </TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> L.T. Hoa, C. Miyazaki, "Bounds on Castelnuovo–Mumford regularity for generalized Cohen–Macaulay graded rings" ''Math. Ann.'' , '''301''' (1995) pp. 587–598 {{MR|1324528}} {{ZBL|0834.13016}} </TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> M.-N. Ishida, "Tsuchihashi's cusp singularities are Buchsbaum singularities" ''Tôhoku Math. J.'' , '''36''' (1984) pp. 191–201 {{MR|742594}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top"> T. Kawasaki, "Local cohomology modules of indecomposable surjective–Buchsbaum modules over Gorenstein local rings" ''J. Math. Soc. Japan'' , '''48''' (1996) pp. 551–566 {{MR|1389995}} {{ZBL|0866.13007}} </TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top"> T. Kawasaki, "Arithmetic Cohen–Macaulayfications of local rings" , ''Proc. 21st Symp. Commutative Algebra in Tokyo, Japan, November 23-26, 1999'' (1999) pp. 88–92</TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top"> Y. Nakamura, "On the Buchsbaum property of associated graded rings" ''J. Algebra'' , '''209''' (1998) pp. 345–366 {{MR|1652142}} {{ZBL|0942.13002}} </TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top"> P. Schenzel, N.V. Trung, N.T. Cuong, "Verallgemeinerte Cohen–Macaulay-Moduln" ''Math. Nachr.'' , '''85''' (1978) pp. 57–73 {{MR|0517641}} {{ZBL|0398.13014}} </TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top"> J. Stückrad, W. Vogel, "Ein Korrekturglied in der Multiplizitätstheorie von D.G. Northcott und Anwendungen" ''Monatsh. Math.'' , '''76''' (1972) pp. 264–271 {{MR|}} {{ZBL|0248.13026}} </TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top"> J. Stückrad, W. Vogel, "Toward a theory of Buchsbaum singularities" ''Amer. J. Math.'' , '''100''' (1978) pp. 727–746 {{MR|0509072}} {{ZBL|0429.14001}} </TD></TR><TR><TD valign="top">[a31]</TD> <TD valign="top"> J. Stückrad, W. Vogel, "Buchsbaum rings and applications" , Springer (1986) {{MR|0881220}} {{MR|0873945}} {{ZBL|0606.13018}} {{ZBL|0606.13017}} </TD></TR><TR><TD valign="top">[a32]</TD> <TD valign="top"> P. Schenzel, "Dualisierende Komplexe in der lokalen Algebra und Buchsbaum–Ringe" , ''Lecture Notes in Mathematics'' , '''907''' , Springer (1982) {{MR|0654151}} {{ZBL|0484.13016}} </TD></TR><TR><TD valign="top">[a33]</TD> <TD valign="top"> P. Schenzel, "On Veronesean embeddings and projections of Veronesean varieties" ''Archiv Math.'' , '''30''' (1978) pp. 391–397 {{MR|0485849}} {{ZBL|0417.14040}} </TD></TR><TR><TD valign="top">[a34]</TD> <TD valign="top"> R.Y. Sharp, "Necessary conditions for the existence of dualizing complexes in commutative algebra" , ''Lecture Notes in Mathematics'' , '''740''' , Springer (1979) pp. 213–229 {{MR|0563505}} {{ZBL|0421.13003}} </TD></TR><TR><TD valign="top">[a35]</TD> <TD valign="top"> Y. Shimoda, "A note on Rees algebras of two-dimensional local domains" ''J. Math. Kyoto Univ.'' , '''19''' (1979) pp. 327–333 {{MR|0545713}} {{ZBL|0447.13010}} </TD></TR><TR><TD valign="top">[a36]</TD> <TD valign="top"> J. Stückrad, "On the Buchsbaum property of Rees and form modules" ''Beitr. Algebra Geom.'' , '''19''' (1985) pp. 83–103 {{MR|0785248}} {{ZBL|0567.13008}} </TD></TR><TR><TD valign="top">[a37]</TD> <TD valign="top"> N.V. Trung, "Toward a theory of generalized Cohen–Macaulay modules" ''Nagoya Math. J.'' , '''102''' (1986) pp. 1–49 {{MR|}} {{ZBL|0649.13008}} {{ZBL|0637.13013}} </TD></TR><TR><TD valign="top">[a38]</TD> <TD valign="top"> N.V. Trung, S. Ikeda, "When is the Rees algebra Cohen–Macaulay?" ''Commun. Algebra'' , '''17''' (1989) pp. 2893–2922 {{MR|}} {{ZBL|0696.13015}} </TD></TR><TR><TD valign="top">[a39]</TD> <TD valign="top"> W. Vasconcelos, "Arithmetic of blowup algebras" , ''London Math. Soc. Lecture Notes'' , '''195''' , Cambridge Univ. Press (1994) {{MR|1275840}} {{ZBL|0813.13008}} </TD></TR><TR><TD valign="top">[a40]</TD> <TD valign="top"> Y. Yoshino, "Maximal Buchsbaum modules of finite projective dimension" ''J. Algebra'' , '''159''' (1993) pp. 240–264 {{MR|1231212}} {{ZBL|0791.13009}} </TD></TR><TR><TD valign="top">[a41]</TD> <TD valign="top"> K. Yamagishi, "The associated graded modules of Buchsbaum modules with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290228.png" />-primary ideals in equi-I-invariant case" ''J. Algebra'' , '''225''' (2000) pp. 1–27 {{MR|1743648}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a42]</TD> <TD valign="top"> K. Yamagishi, "Buchsbaumness in Rees algebras associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130290/b130290229.png" />-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</TD></TR></table> |
Revision as of 16:55, 15 April 2012
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].
References
[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 ![]() |
[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 |
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Buchsbaum ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Buchsbaum_ring&oldid=24390