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A dualizing complex for Stanley–Reisner rings

Published online by Cambridge University Press:  24 October 2008

Hans-Gert Gräbe
Affiliation:
Martin-Luther-Universität Halle-Wittenberg, DDR†

Abstract

In this paper we construct a (multihomogeneous) dualizing complex for A [Δ], the Stanley–Reisner ring of Δ over an arbitrary commutative noetherian ring A with identity, admitting a dualizing complex. In addition, some consequences are discussed.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1]Atiyah, M. F. and Macdonald, I. G.. Introduction to Commutative Algebra (Addison-Wesley, 1969).Google Scholar
[2]Ayomaya, Y. and Goto, S.. On the type of graded Cohen-Macaulay rings. J. Math. Kyoto Univ. 15 (1975), 1923.Google Scholar
[3]Baclawski, K.. Cohen–Macaulay ordered sets. J. Algebra 63 (1980), 226258.CrossRefGoogle Scholar
[4]Baclawski, K. and Garsia, A.. Combinatorial decompositions of a class of rings. Adv. in Math. 39 (1981), 155184.CrossRefGoogle Scholar
[5]Baclawski, K.. Rings with lexicographic straightening law. Adv. in Math. 39 (1981), 185213.CrossRefGoogle Scholar
[6]Björner, A.. Shellable and Cohen—Macaulay partially ordered sets. Trans. Amer. Math. Soc. 260 (1980), 159183.CrossRefGoogle Scholar
[7]BjÖrner, A.. The unimodality conjecture for convex polytopes. Bull. Amer. Math. Soc. (N.S.) 4 (1981), 187189.CrossRefGoogle Scholar
[8]BjÖrner, A. and Wachs, M.. On lexicographically shellable posets. Rep. Dep. Math. Univ. Stockholm 9 (1982).Google Scholar
[9]Foxby, H.-B.. A homological theory of complexes of modules (Kopenhagen University, preprint 19, 1981).Google Scholar
[10]Goto, S. and Watanabe, K.. On graded rings. II. Tokyo J. Math. 1 (1978), 337–261.CrossRefGoogle Scholar
[11]GrÄbe, H.-G.. The canonical module of Stanley-Reisner rings. J. Algebra 86 (1984), 272281.CrossRefGoogle Scholar
[12]GrÄbe, H.-G.. Über den Stanley-Reisner-Ring eines simplizialen Komplexes (Dissertation, Martin-Luther-Univ. Halle/S., 1982).Google Scholar
[13]Hochster, M.. Cohen-Macaulay rings, combinatorics and simplicial complexes. In Ring theory. II. Proceedings of the Second Oklahoma Conference. Lecture Notes in Pure and Appl. Math. vol. 26 (1977).Google Scholar
[14]Matijevic, J. and Roberts, P.. A conjecture of Nagata on graded Cohen-Macaulay modules. J. Math. Kyoto Univ. 14 (1974), 125128.Google Scholar
[15]Reisner, G.. Cohen-Macaulay quotients of polynomial rings. Adv. in Math. 21 (1976), 3049.CrossRefGoogle Scholar
[16]Schenzel, P.. Dualisierende Komplexe in der lokalen Algebra und Buchsbaumringe. Springer Lecture Notes in Math. vol. 907 (1982).CrossRefGoogle Scholar
[17]Spanier, E. H.. Algebraic Topology (McGraw-Hill, 1966).Google Scholar
[18]Stanley, R. P.. Interactions between commutative algebra and combinatorics. Rep. Dep. Math. Univ. Stockholm 4 (1982).Google Scholar
[19]Stanley, R. P.. Cohen-Macaulay complexes. In Higher Combinatorics, ed. Aigner, M. (Reidel, Dortrecht, 1977), pp. 5262.Google Scholar
[20]Stanley, R. P.. The upper bound conjecture and Cohen-Macaulay rings. Stud. Appl. Math. 54 (1975), 135142.CrossRefGoogle Scholar
[21]Stanley, R. P.. The number of faces of a simplicial convex polytope. Adv. in Math. 35 (1980), 236238.CrossRefGoogle Scholar