Differential operators on Stanley-Reisner rings

Author:
J. R. Tripp

Journal:
Trans. Amer. Math. Soc. **349** (1997), 2507-2523

MSC (1991):
Primary 13N10, 16D25, 16P40

DOI:
https://doi.org/10.1090/S0002-9947-97-01749-2

MathSciNet review:
1376559

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an algebraically closed field of characteristic zero, and let be a polynomial ring. Suppose that is an ideal in that may be generated by monomials. We investigate the ring of differential operators on the ring , and , the idealiser of in . We show that and are always right Noetherian rings. If is a square-free monomial ideal then we also identify all the two-sided ideals of . To each simplicial complex on there is a corresponding square-free monomial ideal , and the Stanley-Reisner ring associated to is defined to be . We find necessary and sufficient conditions on for to be left Noetherian.

**[1]**W. C. Brown,*A Note on Higher Derivations and Ordinary Points of Curves*, Rocky Mountain J. Math.**14**(1984), 397-402. MR**85k:13007****[2]**J. N. Bernstein, I. M. Gelfand and S. I. Gelfand,*Differential Operators on the Cubic Cone*, Russian Math. Surveys**27**(1972), 169-174. MR**52:6024****[3]**P. Brumatti and A. Simis,*The Module of Derivations of a Stanley-Reisner Ring*, Proc. Amer. Math. Soc.**123**(1995), 1309-1318. MR**95f:13014****[4]**W. Bruns and J. Herzog,*Cohen-Macaulay Rings*, Cambridge University Press, 1993. MR**95h:13020****[5]**S. C. Coutinho and M. P. Holland,*Noetherian -Bimodules*, J. Alg.**168**(1994), 434-443. MR**95i:16025****[6]**J. C. McConnell and J. C. Robson,*Noncommutative Noetherian Rings*, Wiley, 1987. MR**89j:16023****[7]**G. Müller,*Lie Algebras Attached to Simplicial Complexes*, J. Alg.**177**(1995), 132-141. CMP**1996:3**.**[8]**J. L. Muhasky,*The Differential Operator Ring of an Affine Curve*, Trans. Amer. Math. Soc.**307**(1988), 705-723. MR**89i:16001****[9]**I. M. Musson,*Rings of Differential Operators and Zero Divisors*, J. Alg.**125**(1989), 489-501. MR**90k:16014****[10]**G. A. Reisner,*Cohen-Macaulay Quotients of Polynomial Rings*, Advances in Math.**21**(1976), 30-49. MR**53:10819****[11]**R. Resco,*Affine Domains of Finite Gelfand-Kirillov Dimension which are Right, but not Left, Noetherian*, Bull. London Math. Soc.**16**(1984), 590-594. MR**85k:16047****[12]**J. C. Robson,*Idealizers and Hereditary Noetherian Prime Rings*, J. Algebra**22**(1972), 45-81. MR**45:8687****[13]**J. C. Robson and L. W. Small,*Orders Equivalent to the First Weyl Algebra*, Quart. J. Math. Oxford (2)**37**(1986), 475-482. MR**88a:16010****[14]**S. P. Smith and J. T. Stafford,*Differential Operators on an Affine Curve*, Proc. London Math. Soc. (3)**56**(1988), 229-259. MR**89d:14039**

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Additional Information

**J. R. Tripp**

Affiliation:
Pure Math Section, Hicks Building, University of Sheffield, Sheffield S3 7RH, England

Email:
J.R.Tripp@Sheffield.ac.uk

DOI:
https://doi.org/10.1090/S0002-9947-97-01749-2

Received by editor(s):
May 9, 1995

Received by editor(s) in revised form:
January 5, 1996

Additional Notes:
I should like to thank Martin Holland for numerous helpful discussions, the referee for constructive comments, and the EPSRC for their funding while this work was completed.

Article copyright:
© Copyright 1997
American Mathematical Society