Buchsbaum ring

In mathematics, Buchsbaum rings are Noetherian local rings such that every system of parameters is a weak sequence. A sequence ${\displaystyle (a_{1},\cdots ,a_{r})}$ of the maximal ideal $m$ is called a weak sequence if ${\displaystyle m\cdot ((a_{1},\cdots ,a_{i-1})\colon a_{i})\subset (a_{1},\cdots ,a_{i-1})}$ for all $i$.

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.

References

• Buchsbaum, D. (1966), "Complexes in local ring theory", in Herstein, I. N., Some aspects of ring theory, Centro Internazionale Matematico Estivo (C.I.M.E.). II Ciclo, Varenna (Como), 23-31 agosto, 1965, Rome: Edizioni cremonese, pp. 223–228, ISBN 978-3-642-11035-1, MR 0223393
• Goto (, Shiro (2001), "Buchsbaum_ring", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• 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, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16844-7, MR 881220