Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $k$ be an algebraically closed field of characteristic zero, and let $R=k[x_{1},\dots ,x_{n}]$ be a polynomial ring. Suppose that $I$ is an ideal in $R$ that may be generated by monomials. We investigate the ring of differential operators $\mathcal {D}(R/I)$ on the ring $R/I$, and $\mathcal {I}_{R}(I)$, the idealiser of $I$ in $R$. We show that $\mathcal {D}(R/I)$ and $\mathcal {I}_{R}(I)$ are always right Noetherian rings. If $I$ is a square-free monomial ideal then we also identify all the two-sided ideals of $\mathcal {I}_{R}(I)$. To each simplicial complex $\Delta $ on $V=\{v_{1},\dots ,v_{n}\}$ there is a corresponding square-free monomial ideal $I_{\Delta }$, and the Stanley-Reisner ring associated to $\Delta $ is defined to be $k[\Delta ]=R/I_{\Delta }$. We find necessary and sufficient conditions on $\Delta $ for $\mathcal {D}(k[\Delta ])$ to be left Noetherian.


References [Enhancements On Off] (What's this?)

  • [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 $\mathcal {D}$-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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 13N10, 16D25, 16P40

Retrieve articles in all journals with MSC (1991): 13N10, 16D25, 16P40


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

American Mathematical Society