In mathematics , Buchsbaum rings are Noetherian local rings such that every system of parameters is a weak sequence.
A sequence
(
a
1
,
⋯ ⋯ -->
,
a
r
)
{\displaystyle (a_{1},\cdots ,a_{r})}
m
{\displaystyle m}
m
⋅ ⋅ -->
(
(
a
1
,
⋯ ⋯ -->
,
a
i
− − -->
1
)
: : -->
a
i
)
⊂ ⊂ -->
(
a
1
,
⋯ ⋯ -->
,
a
i
− − -->
1
)
{\displaystyle m\cdot ((a_{1},\cdots ,a_{i-1})\colon a_{i})\subset (a_{1},\cdots ,a_{i-1})}
i
{\displaystyle i}
They were introduced by Jürgen Stückrad and Wolfgang Vogel (1973 ) and are named after David Buchsbaum .
Every Cohen–Macaulay local ring is a Buchsbaum ring. Every Buchsbaum ring is a generalized Cohen–Macaulay ring .
References
Buchsbaum, D. (1966), "Complexes in local ring theory", in Herstein, I. N. (ed.), Some aspects of ring theory ISBN 978-3-642-11035-1 MR 0223393 Goto, Shiro (2001) [1994], "Buchsbaum ring" , Encyclopedia of Mathematics EMS Press Stückrad, Jürgen; Vogel, Wolfgang (1973), "Eine Verallgemeinerung der Cohen-Macaulay Ringe und Anwendungen auf ein Problem der Multiplizitätstheorie" , Journal of Mathematics of Kyoto University , 13 : 513–528, ISSN 0023-608X , MR 0335504 Stückrad, Jürgen; Vogel, Wolfgang (1986), Buchsbaum rings and applications Springer-Verlag , ISBN 978-3-540-16844-7 MR 0881220
