Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-30T19:58:11.132Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiplicities in graded rings II: integral equivalence and the Buchsbaum–Rim multiplicity

Published online by Cambridge University Press:  24 October 2008

D. Kirby  and
D. Rees
D. Kirby
Affiliation:
The University, Southampton SO9 5NH
D. Rees
Affiliation:
6 Hillcrest Park, Exeter EX4 4SH, Devon
Get access Rights & Permissions [Opens in a new window]

Extract

While this paper is principally a continuation of [5], with as its object the application of sections 6 and 7 of that paper to obtain results related to the Buchsbaum–Rim multiplicity, it also has connections with [8] which are the subject of the first of the four sections. These concern integral equivalence of finitely generated R-modules. where R is an arbitrary noetherian ring. We therefore introduce a finitely generated R-module M and relate to it a short exact sequence (s.e.s.),

where F is a free module generated by m elements u1,…, um, and L is generated by elements yj, (j = 1, …, n), of F. We identify the elements u1, …, um with a set of indeterminates X1, …, Xm, and F with the R-module S1 of elements of degree 1 of the graded ring S = R[X1, …, Xm].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 3 , April 1996 , pp. 425 - 445
Copyright
Copyright © Cambridge Philosophical Society 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Buchsbaum, D. A. and Rim, D. S.. A generalised Koszul complex II. Depth and multiplicity. Trans. Amer. Mutti. Soc. 111 (1964), 197–224.CrossRefGoogle Scholar
[2]Kaplansky, I.. fields and rings (University of Chicago Press, 1969).Google Scholar
[3]Kirby, D.. On the Buchsbaum–Rim multiplicity associated with a Matrix. J. Land. Math. Soc. (2), 32 (1985), 57–61.Google Scholar
[4]Kirby, D.. Graded multiplicity theory and Hubert functions. J. Land. Math. Soc. (2), 36 (1987), 16–22.Google Scholar
[5]Kirby, D. and Rees, D.. Multiplicities in graded rings I: the general theory, Algebra: Syzygies, Multiplicities and birational Algebra (Heinzer, William J., Hunecke, Craig L. and Sally, Judith D., eds.). vol. 159, Contemporary Mathematics. 1994.Google Scholar
[6]Rees, D.. Generalisations of reductions and mixed multiplicities. J. Land. Math. Soc. (2), 29 (1984), 397–414.Google Scholar
[7]Rees, D.. Amao's theorem and reduction criteria. J. Lond. Math. Soc. (2), 32 (1985), 404–410.CrossRefGoogle Scholar
[8]Rees, D.. Reductions of modules. Math. Proc. Camb. Phil. Soc. 101 (1987), 431–449.Google Scholar
[9]Rees, D.. Lectures on the asymptotic theory of ideals. L.M.S. Lecture Note Series, vol. 113 (Cambridge University Press, 1988).Google Scholar
[10]Rees, D. and Sally, J. D.. General elements and joint reductions. Michigan Math. J. 35 (1988), 241–254.Google Scholar
[11]Kleiman, S. and Thorup, A.. A geometric theory of the Buchsbaum–Rim multiplicity. Jour. of Alg. 167 (1994), 168–231.Google Scholar
[12]Kleiman, S. and Thoeup, A.. Mixed Buchsbaum–Rim multiplicities. Preprint Series of Math. Inst., Copenhagen University, no. 31 (1994). 1–36.Google Scholar