Differential operators on Stanley-Reisner rings
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- by J. R. Tripp PDF
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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.References
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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
- 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.
- © Copyright 1997 American Mathematical Society
- 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