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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
MathSciNet review: 1376559
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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.

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Additional Information

J. R. Tripp
Affiliation: Pure Math Section, Hicks Building, University of Sheffield, Sheffield S3 7RH, England

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