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Differential Operators on Stanley-Reisner Rings

J. R. Tripp
Transactions of the American Mathematical Society
Vol. 349, No. 6 (Jun., 1997), pp. 2507-2523
Stable URL: http://www.jstor.org/stable/2155521
Page Count: 17
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Differential Operators on Stanley-Reisner Rings
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Abstract

Let k be an algebraically closed field of characteristic zero, and let R = k [ x1, ..., xn ] 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 D(R/I) on the ring R/I, and IR(I), the idealiser of I in R. We show that D(R/I) and IR(I) are always right Noetherian rings. If I is a square-free monomial ideal then we also identify all the two-sided ideals of IR(I). To each simplicial complex Δ on V = {v1, ..., vn} there is a corresponding square-free monomial ideal IΔ, and the Stanley-Reisner ring associated to Δ is defined to be k [ Δ ] = R/IΔ. We find necessary and sufficient conditions on Δ for D (k [ Δ ]) to be left Noetherian.

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